\section{Introduction}

This document is the first iteration of an attempt to collate information about small-rank groups of Lie type over small fields, and their representation theory over the defining field. This information is important in the author's work on subgroup structure of exceptional groups of Lie type.

The most important information in that work is information about $\Ext^1$ between simple modules, and so in Version 1 of this document, that data is almost all of the data available. In addition, a lot of information about the dimensions of the simple and Weyl modules is included.

More generally, one may expect to include details about the socle structure of the projective modules, Jordan block structure of the action of unipotent elements, decompositions of symmetric and exterior powers of simple modules, and tensor products of modules, traces of semisimple elements and so on.

The ideal place for such information is a dedicated website, connected to a database that could be queried to produce the information required. This document, while imperfect, will have to suffice for now.

\medskip

Note that (almost) all of these data were constructed via computer and with Magma. There are various articles that compute $\Ext^1$ between simple modules for low-rank Lie type groups over small fields, but given that long computational papers are notoriously error prone, the author felt it best to compute the table here independently, so that we now have two sources for (some of) these tables.

The labelling conventions for our simple modules follow the author's recent papers on maximal subgroups, with a dictionary between this and highest-weight labellings being given at the start of each section.

\newpage

\tableofcontents

\newpage
\section{\texorpdfstring{$\SL_3(2)$}{SL(3,2)}}

\subsection*{Block information}

\begin{center}\begin{tabular}{ccc}
\hline Block & Kernel & Modules
\\\hline Block 1 & 1 & $1$, $3$, $3^*$
\\Block 2 & 1 & $8$
\\\hline\end{tabular}\end{center}

\subsection*{Highest weight labels}

\begin{center}\begin{tabular}{ccccc}
\hline $\lambda$&00&10&01&11
\\\hline $L(\lambda)$ &$1$&$3$&$3^*$&$8$\\ $\dim(W(\lambda))$&$1$&$3$&$3$&$8$
\\\hline
\end{tabular}\end{center}

\subsection*{Extension information}

\begin{center}\begin{tabular}{c|ccc}
Block 1&$1$&$3$&$3^*$
\\\hline $1$&$0$&$1$&$1$
\\ $3$&$1$&$0$&$1$
\\ $3^*$&$1$&$1$&$0$
\\\end{tabular}\end{center}

\newpage
\section{\texorpdfstring{$\SL_3(3)$}{SL(3,3)}}

\subsection*{Block information}

\begin{center}\begin{tabular}{ccc}
\hline Block & Kernel & Modules
\\\hline Block 1 & 1 & $1$, $3$, $3^*$, $6$, $6^*$, $7$, $15$, $15^*$
\\Block 2 & 1 & $27$
\\\hline\end{tabular}\end{center}

\subsection*{Identification of modules}

\begin{enumerate}
\item Fix $3$ and $3^*$.
\item $6=S^2(3^*)$.
\item $15=\Lambda^2(6^*)$.
\end{enumerate}

\subsection*{Highest weight labels}

\begin{center}\begin{tabular}{cccccccccc}
\hline $\lambda$&00&10&01&02&20&11&21&12&22
\\\hline $L(\lambda)$ &$1$&$3$&$3^*$&$6$&$6^*$&$7$&$15$&$15^*$&$27$\\ $\dim(W(\lambda))$&$1$&$3$&$3$&$6$&$6$&$8$&$15$&$15$&$27$
\\\hline
\end{tabular}\end{center}


\subsection*{Extension information}

\begin{center}\begin{tabular}{c|cccccccc}
Block 1&$1$&$3$&$3^*$&$6$&$6^*$&$7$&$15$&$15^*$
\\\hline $1$&$0$&$0$&$0$&$0$&$0$&$1$&$1$&$1$
\\ $3$&$0$&$0$&$0$&$0$&$1$&$1$&$0$&$0$
\\ $3^*$&$0$&$0$&$0$&$1$&$0$&$1$&$0$&$0$
\\ $6$&$0$&$0$&$1$&$0$&$1$&$0$&$0$&$0$
\\ $6^*$&$0$&$1$&$0$&$1$&$0$&$0$&$0$&$0$
\\ $7$&$1$&$1$&$1$&$0$&$0$&$0$&$0$&$0$
\\ $15$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\ $15^*$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\end{tabular}\end{center}

\newpage
\section{\texorpdfstring{$\SL_3(4)$}{SL(3,4)}}

\subsection*{Block information}

\begin{center}\begin{tabular}{ccc}
\hline Block & Kernel & Modules
\\ \hline 1 & 3 & $8_1$, $8_2$, $9_{1,2}$, $9_{1,2}^*$
\\ 2 & 3 & $64$
\\ 3 & 1 & $3_1$, $3_2^*$, $\bar 9_{1,2}^*$ $24_{1,2}$, $24_{2,1}^*$
\\ 4 & 1 & Dual to block 3
\\\hline\end{tabular}\end{center}

\subsection*{Identification of modules}

We write $3_2$ for the twist of $3_1$, so that that $3_1$ and $3_2$ restrict to the same module for $\SL_3(2)$. Write $8_1$ for the module such that $8_1\otimes 3_2$ is irreducible.
\begin{enumerate}
\item $9_{1,2}=3_1\otimes 3_2$
\item $\bar 9_{1,2}=3_1\otimes 3_2^*$
\item $24_{1,2}=3_1\otimes 8_2$ and $24_{2,1}=3_2\otimes 8_1$
\end{enumerate}

\subsection*{Extension information}

\begin{center}\begin{tabular}{c|ccccc}
Block 1 &$1$&$8_1$&$8_2$&$9_{1,2}$&$9_{1,2}^*$
\\\hline $1$&$0$&$0$&$0$&$2$&$2$
\\ $8_1$&$0$&$0$&$0$&$1$&$1$
\\ $8_2$&$0$&$0$&$0$&$1$&$1$
\\ $9_{1,2}$&$2$&$1$&$1$&$0$&$0$
\\ $9_{1,2}^*$&$2$&$1$&$1$&$0$&$0$
\end{tabular}\end{center}

\begin{center}\begin{tabular}{c|ccccc}
Block 3 &$3_1$&$3_2^*$&$\bar 9_{1,2}^*$&$24_{1,2}$&$24_{2,1}^*$
\\\hline $3_1$&$0$&$2$&$1$&$0$&$1$
\\ $3_2^*$&$2$&$0$&$1$&$1$&$0$
\\ $\bar 9_{1,2}^*$&$1$&$1$&$0$&$0$&$0$
\\ $24_{1,2}$&$0$&$1$&$0$&$0$&$0$
\\ $24_{2,1}^*$&$1$&$0$&$0$&$0$&$0$
\end{tabular}\end{center}


\newpage
\section{\texorpdfstring{$\SL_3(5)$}{SL(3,5)}}

\subsection*{Block information}

\begin{center}\begin{tabular}{ccc}
\hline Block & Kernel & Modules
\\\hline Block 1 & $1$ & $1$, $3$, $3^*$, $6$, $6^*$, $8$, $10$, $10^*$, $15_1$, $15_1^*$, $15_2$, $15_2^*$
\\ && $18$, $18^*$, $19$, $35$, $35^*$, $39$, $39^*$, $60$, $60^*$, $63$, $90$, $90^*$
\\Block 2 & $1$ & $125$
\\\hline\end{tabular}\end{center}

\subsection*{Identification of modules}

\begin{enumerate}
\item Fix $3$ and $3^*$.
\item $6=S^2(3^*)$.
\item $10$ is a submodule of $3^*\otimes 6$.
\item $15_1$ is a submodule of $S^2(6^*)$.
\item $15_2=\Lambda^2(6^*)$.
\item $18$ is a submodule of $6\otimes 10$.
\item $35$ is a submodule of $\Lambda^2(10^*)$.
\item $39$ is a submodule of $\Lambda^2(15_1^*)$.
\item $60$ is a submodule of $6\otimes 19$.
\item $90$ is a submodule of $\Lambda^2(18^*)$.
\end{enumerate}

\subsection*{Highest weight labels}

\begin{center}\begin{tabular}{ccccccccccccccc}
\hline $\lambda$&00&10&01&02&20&11&03&30&40&04&21&12&13&31
\\\hline $L(\lambda)$ &$1$&$3$&$3^*$&$6$&$6^*$&$8$&$10$&$10^*$&$15_1$&$15_1^*$&$15_2$&$15_2^*$&$18$&$18^*$
\\ $\dim(W(\lambda))$&$1$&$3$&$3$&$6$&$6$&$8$&$10$&$10$&$15$&$15$&$15$&$15$&$24$&$24$
\\\hline
\end{tabular}\end{center}

\begin{center}\begin{tabular}{cccccccccccc}
\hline $\lambda$&22&41&14&32&23&24&42&33&43&34&44
\\\hline $L(\lambda)$ &$19$&$35$&$35^*$&$39$&$39^*$&$60$&$60^*$&$63$&$90$&$90^*$&$125$
\\ $\dim(W(\lambda))$&$27$&$35$&$35$&$42$&$42$&$60$&$60$&$64$&$90$&$90$&$125$
\\\hline
\end{tabular}\end{center}

\subsection*{Extension information}

\begin{center}\begin{footnotesize}\begin{tabular}{c|ccccccccccccccccccccc}
&$1$&$3$&$3^*$&$6$&$6^*$&$8$&$10$&$10^*$&$15_1$&$15_1^*$&
$15_2$&$15_2^*$&$18$&$18^*$&$19$&$35$&$35^*$&$39$&$60$&$63$&$90$
\\\hline $1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$1$&$0$&$1$&$0$
\\ $3$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$1$&$1$&$0$&$0$&$1$&$0$&$1$&$0$
\\ $3^*$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$1$&$0$&$1$&$0$&$0$&$1$&$0$&$1$&$0$
\\ $6$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$1$&$0$&$1$&$1$&$0$&$0$&$0$&$0$&$0$
\\ $6^*$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$1$&$1$&$0$&$1$&$1$&$0$&$0$&$1$
\\ $8$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$1$&$1$&$1$&$0$&$0$&$1$&$1$&$0$&$0$
\\ $10$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$1$&$0$&$0$&$0$&$0$&$1$&$0$&$0$
\\ $10^*$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\ $15_1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$1$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\ $15_1^*$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$1$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\ $15_2$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$1$&$0$&$0$&$0$&$1$&$0$
\\ $15_2^*$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$1$&$1$&$0$&$1$&$0$
\\ $18$&$0$&$0$&$1$&$1$&$0$&$1$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\ $18^*$&$0$&$1$&$0$&$0$&$1$&$1$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\ $19$&$0$&$1$&$1$&$1$&$1$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\ $35$&$0$&$0$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\ $35^*$&$0$&$0$&$0$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\ $39$&$1$&$1$&$1$&$0$&$1$&$1$&$0$&$0$&$0$&$0$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\ $39^*$&$1$&$1$&$1$&$1$&$0$&$1$&$0$&$0$&$0$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\ $60$&$0$&$0$&$0$&$0$&$0$&$1$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\ $60^*$&$0$&$0$&$0$&$0$&$0$&$1$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\ $63$&$1$&$1$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$1$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\ $90$&$0$&$0$&$0$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\ $90^*$&$0$&$0$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\end{tabular}\end{footnotesize}\end{center}


\newpage
\section{\texorpdfstring{$\SL_3(7)$}{SL(3,7)}}

\subsection*{Block information}

\begin{center}\begin{tabular}{ccc}
\hline Block & Kernel & Modules
\\\hline Block 1 & $3$ & $1$, $8$, $10$, $10^*$, $27$, $28$, $28^*$, $35$, $35^*$, $37$, $71$, $71^*$, $117$, $154$, $154^*$, $215$
\\Block 2 & $3$ & $343$
\\ Block 3 & $1$ & $3$, $6$, $15_1$, $15_2$, $21$, $24$, $33$, $36$, $42$, $63$, $75$, $105$, $114$, $162$, $210$, $273$
\\ Block 4 & $1$ & Dual to block 3
\\\hline\end{tabular}\end{center}

\subsection*{Identification of modules}

\begin{enumerate}
\item Fix $3$ and $3^*$. This fixes all modules in Blocks 3 and 4 but $15_1$ and $15_2$.
\item $10$ is a submodule of $3^*\otimes 6$.
\item $15_1$ is a submodule of $S^2(6^*)$.
\item $15_2=\Lambda^2(6^*)$.
\item $28$ is a submodule of $S^2(10^*)$.
\item $35$ is a submodule of $\Lambda^2(10^*)$.
\item $71$ is a submodule of $8\otimes 28^*$.
\item $154$ is a submodule of $10\otimes 37$.
\end{enumerate}

\subsection*{Highest weight labels}

\begin{center}\begin{tabular}{cccccccccccccccccc}
\hline $\lambda$&00&11&03&30&22&60&06&41&14&33&25&52&44&36&63&55&66
\\\hline $L(\lambda)$ &$1$&$8$&$10$&$10^*$&$27$&$28$&$28^*$&$35$&$35^*$&$37$&$71$&$71^*$&$117$&$154$&$154^*$&$215$&$343$
\\ $\dim(W(\lambda))$&$1$&$8$&$10$&$10$&$27$&$28$&$28$&$35$&$35$&$64$&$81$&$81$&$125$&$154$&$154$&$216$&$343$
\\\hline
\end{tabular}\end{center}

\begin{center}\begin{tabular}{ccccccccccccccccc}
\hline $\lambda$&10&02&40&21&05&13&51&24&32&16&43&62&35&54&46&65
\\\hline $L(\lambda)$ &$3$&$6$&$15_1$&$15_2$&$21$&$24$&$33$&$36$&$42$&$63$&$75$&$105$&$114$&$162$&$210$&$273$
\\ $\dim(W(\lambda))$&$3$&$6$&$15$&$15$&$21$&$24$&$48$&$60$&$42$&$63$&$90$&$105$&$120$&$165$&$210$&$273$
\\\hline
\end{tabular}\end{center}

\subsection*{Extension information}

\begin{center}\begin{tabular}{c|cccccccccccccccc}
Block 1&$1$&$8$&$10$&$10^*$&$27$&$28$&$28^*$&$35$&$35^*$&$37$&$71$&$71^*$&$117$&$154$&$154^*$&$215$
\\ \hline $1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$1$&$1$&$0$&$0$&$0$&$1$
\\ $8$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$2$&$1$&$1$&$3$&$0$&$0$&$0$
\\ $10$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$1$&$1$&$1$&$1$&$0$&$0$&$1$
\\ $10^*$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$1$&$1$&$1$&$1$&$0$&$0$&$1$
\\$27$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$1$&$1$&$1$&$2$&$1$&$1$&$0$
\\$28$&$0$&$0$&$0$&$0$&$0$&$0$&$1$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\$28^*$&$0$&$0$&$0$&$0$&$0$&$1$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\$35$&$0$&$0$&$0$&$0$&$0$&$0$&$1$&$0$&$0$&$0$&$0$&$1$&$0$&$0$&$1$&$1$
\\ $35^*$&$0$&$0$&$0$&$0$&$0$&$1$&$0$&$0$&$0$&$0$&$1$&$0$&$0$&$1$&$0$&$1$
\\ $37$&$0$&$2$&$1$&$1$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\ $71$&$1$&$1$&$1$&$1$&$1$&$0$&$0$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\ $71^*$&$1$&$1$&$1$&$1$&$1$&$0$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\ $117$&$0$&$3$&$1$&$1$&$2$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\ $154$&$0$&$0$&$0$&$0$&$1$&$0$&$0$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\ $154^*$&$0$&$0$&$0$&$0$&$1$&$0$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\ $215$&$1$&$0$&$1$&$1$&$0$&$0$&$0$&$1$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\end{tabular}\end{center}


\begin{center}\begin{tabular}{c|cccccccccccccccc}
Block 3&$3$&$6$&$15_1$&$15_2$&$21$&$24$&$33$&$36$&$42$&$63$&$75$&$105$&$114$&$162$&$210$&$273$
\\\hline $3$&$0$&$0$&$0$&$0$&$0$&$0$&$1$&$1$&$0$&$0$&$1$&$0$&$1$&$1$&$0$&$0$
\\ $6$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$1$&$0$&$0$&$2$&$0$&$1$&$1$&$0$&$0$
\\ $15_1$&$0$&$0$&$0$&$0$&$0$&$0$&$1$&$1$&$0$&$1$&$0$&$0$&$0$&$1$&$0$&$1$
\\ $15_2$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$1$&$0$&$0$&$3$&$0$&$2$&$1$&$0$&$0$
\\ $21$&$0$&$0$&$0$&$0$&$0$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$1$&$0$
\\ $24$&$0$&$0$&$0$&$0$&$0$&$0$&$1$&$1$&$0$&$0$&$1$&$1$&$1$&$1$&$1$&$0$
\\ $33$&$1$&$0$&$1$&$0$&$1$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\ $36$&$1$&$1$&$1$&$1$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\ $42$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$1$&$0$&$1$&$1$&$1$&$0$&$0$
\\ $63$&$0$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\ $75$&$1$&$2$&$0$&$3$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\$105$&$0$&$0$&$0$&$0$&$0$&$1$&$0$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\$114$&$1$&$1$&$0$&$2$&$0$&$1$&$0$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\$162$&$1$&$1$&$1$&$1$&$0$&$1$&$0$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\$210$&$0$&$0$&$0$&$0$&$1$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\$273$&$0$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\end{tabular}\end{center}



\newpage
\section{\texorpdfstring{$\SU_3(3)$}{SU(3,3)}}


\subsection*{Block information}

\begin{center}\begin{tabular}{ccc}
\hline Block & Kernel & Modules
\\\hline Block 1 & $1$ & $1$, $3$, $3^*$, $6$, $6^*$, $7$, $15$, $15^*$
\\Block 2 & $1$ & $27$
\\\hline\end{tabular}\end{center}

\subsection*{Identification of modules}

\begin{enumerate}
\item Fix $3$ and $3^*$.
\item $6=S^2(3^*)$.
\item $15=\Lambda^2(6^*)$.
\end{enumerate}

\subsection*{Extension information}

\begin{center}\begin{tabular}{c|cccccccc}
Block 1&$1$&$3$&$3^*$&$6$&$6^*$&$7$&$15$&$15^*$
\\\hline $1$&$0$&$0$&$0$&$1$&$1$&$1$&$0$&$0$
\\ $3$&$0$&$0$&$1$&$0$&$0$&$1$&$0$&$1$
\\ $3^*$&$0$&$1$&$0$&$0$&$0$&$1$&$1$&$0$
\\ $6$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$1$
\\ $6^*$&$1$&$0$&$0$&$0$&$0$&$0$&$1$&$0$
\\ $7$&$1$&$1$&$1$&$0$&$0$&$0$&$0$&$0$
\\ $15$&$0$&$0$&$1$&$0$&$1$&$0$&$0$&$0$
\\ $15^*$&$0$&$1$&$0$&$1$&$0$&$0$&$0$&$0$
\end{tabular}\end{center}

\newpage
\section{\texorpdfstring{$\SU_3(4)$}{SU(3,4)}}


\subsection*{Block information}

\begin{center}\begin{tabular}{ccc}
\hline Block & Kernel & Modules
\\ \hline 1 & 1 & $1$, $3_1$, $3_1^*$, $3_2$, $3_2^*$, $8_1$, $8_2$, $9_{1,2}$, $9_{1,2}^*$, $\bar 9_{1,2}$, $\bar 9_{1,2}^*$, $24_{1,2}$, $24_{1,2}^*$, $24_{2,1}$, $24_{2,1}^*$
\\ 2 & 1 & $64$
\\\hline\end{tabular}\end{center}

\subsection*{Identification of modules}

We write $3_2$ for the twist of $3_1$, so that that $3_1$ and $3_2$ restrict to the same module for $\SL_3(2)$. Write $8_1$ for the module such that $8_1\otimes 3_2$ is irreducible.
\begin{enumerate}
\item $9_{1,2}=3_1\otimes 3_2$
\item $\bar 9_{1,2}=3_1\otimes 3_2^*$
\item $24_{1,2}=3_1\otimes 8_2$ and $24_{2,1}=3_2\otimes 8_1$
\end{enumerate}


\subsection*{Extension information}

\begin{center}\begin{tabular}{c|ccccccccccccccc}
Block 1& $1$&$3_1$&$3_1^*$&$3_2$&$3_2^*$&$8_1$&$8_2$&$9_{1,2}$&$9_{1,2}^*$&$\bar 9_{1,2}$&$\bar 9_{1,2}^*$&$24_{1,2}$&$24_{1,2}^*$&$24_{2,1}$&$24_{2,1}^*$
\\\hline $1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$1$&$1$&$1$&$1$&$0$&$0$&$0$&$0$
\\ $3_1$&$0$&$0$&$0$&$1$&$1$&$0$&$0$&$0$&$1$&$0$&$0$&$0$&$0$&$1$&$0$
\\$3_1^*$&$0$&$0$&$0$&$1$&$1$&$0$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$1$
\\ $3_2$&$0$&$1$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$1$&$0$&$0$&$1$&$0$&$0$
\\ $3_2^*$&$0$&$1$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$1$&$1$&$0$&$0$&$0$
\\ $8_1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$1$&$1$&$0$&$0$&$0$&$0$
\\ $8_2$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$1$&$1$&$0$&$0$&$0$&$0$&$0$&$0$
\\ $9_{1,2}$&$1$&$0$&$1$&$0$&$0$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\ $9_{1,2}^*$&$1$&$1$&$0$&$0$&$0$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\ $\bar 9_{1,2}$&$1$&$0$&$0$&$1$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\ $\bar 9_{1,2}^*$&$1$&$0$&$0$&$0$&$1$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\ $24_{1,2}$&$0$&$0$&$0$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\ $24_{1,2}^*$&$0$&$0$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\ $24_{2,1}$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\ $24_{2,1}^*$&$0$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\end{tabular}\end{center}

\newpage
\section{\texorpdfstring{$\SU_3(5)$}{SU(3,5)}}

\subsection*{Block information}

\begin{center}\begin{tabular}{ccc}
\hline Block & Kernel & Modules
\\\hline 1 & 3 & $1$, $8$, $10$, $10^*$, $19$, $35$, $35^*$, $63$
\\ 2 & 3 & $125$
\\ 3 & 1 & $3$, $6$, $15_1$, $15_2$, $18$, $39$, $60$, $90$
\\ 4 & 1 & Dual to block 3
\\\hline\end{tabular}\end{center}

\subsection*{Identification of modules}

\begin{enumerate}
\item Fix $3$ and $3^*$. This fixes all modules in blocks 3 and 4 except for $15_1$ and $15_2$.
\item $10$ is a submodule of $3^*\otimes 6$.
\item $15_1$ is a submodule of $S^2(6^*)$.
\item $15_2=\Lambda^2(6^*)$.
\item $35$ is a submodule of $\Lambda^2(10^*)$.
\end{enumerate}

\subsection*{Extension information}

\begin{center}\begin{tabular}{c|cccccccc}
Block 1&$1$&$8$&$10$&$10^*$&$19$&$35$&$35^*$&$63$
\\ \hline $1$&$0$&$0$&$0$&$0$&$2$&$0$&$0$&$1$
\\ $8$&$0$&$0$&$0$&$0$&$3$&$1$&$1$&$2$
\\ $10$&$0$&$0$&$0$&$1$&$0$&$1$&$0$&$1$
\\ $10^*$&$0$&$0$&$1$&$0$&$0$&$0$&$1$&$1$
\\ $19$&$2$&$3$&$0$&$0$&$0$&$0$&$0$&$0$
\\ $35$&$0$&$1$&$1$&$0$&$0$&$0$&$0$&$0$
\\ $35^*$&$0$&$1$&$0$&$1$&$0$&$0$&$0$&$0$
\\ $63$&$1$&$2$&$1$&$1$&$0$&$0$&$0$&$0$
\end{tabular}\end{center}

\begin{center}\begin{tabular}{c|cccccccc}
Block 3&$3$&$6$&$15_1$&$15_2$&$18$&$39$&$60$&$90$
\\ \hline $3$&$0$&$0$&$0$&$0$&$2$&$3$&$0$&$0$
\\ $6$&$0$&$0$&$1$&$0$&$1$&$2$&$1$&$0$
\\ $15_1$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$1$
\\ $15_2$&$0$&$0$&$0$&$0$&$1$&$1$&$1$&$1$
\\ $18$&$2$&$1$&$0$&$1$&$0$&$0$&$0$&$0$
\\ $39$&$3$&$2$&$0$&$1$&$0$&$0$&$0$&$0$
\\ $60$&$0$&$1$&$0$&$1$&$0$&$0$&$0$&$0$
\\ $90$&$0$&$0$&$1$&$1$&$0$&$0$&$0$&$0$
\end{tabular}\end{center}


\newpage
\section{\texorpdfstring{$\SU_3(7)$}{SU(3,7)}}

\subsection*{Block information}

\begin{center}\begin{tabular}{ccc}
\hline Block & Kernel & Modules
\\\hline 1 & 1 & $1$, $3$, $3^*$, $6$, $6^*$, $8$, $10$, $10^*$, $15_1$, $15_1^*$, $15_2$, $15_2^*$, $21$, $21^*$, $24$, $24^*$, $27$, $28$, $28^*$,
\\ &&$33$, $33^*$, $35$, $35^*$, $36$, $36^*$, $37$, $42$, $42^*$, $63$, $63^*$, $71$, $71^*$, $75$, $75^*$, $105$, $105^*$,
\\ && $114$, $114^*$, $117$, $154$, $154^*$, $162$, $162^*$, $210$, $210^*$, $215$, $273$, $273^*$
\\ 2 & 1 & $343$
\\\hline\end{tabular}\end{center}

\subsection*{Identification of modules}

We produce an identification here that is consistent with the choice of highest weight labels from $\SL_3(7)$.

\begin{enumerate}
\item Fix $3$, $3^*$.
\item $6=S^2(3^*)$.
\item $10\mid 3^*\otimes 6$.
\item $15_1\mid S^2(6^*)$.
\item $15_2=\Lambda^2(6^*)$.
\item $21\oplus 24=3^*\otimes 15_1^*$.
\item $28\mid S^2(10^*)$.
\item $33$ is a factor of $3^*\otimes 21^*$.
\item $35\mid \Lambda^2(10^*)$.
\item $36 \subset S^2(21^*)$.
\item $42\oplus 63\mid \Lambda^2(15_1^*)$.
\item $71$ is a submodule of $8\otimes 28^*$.
\item $75$ is a factor of $6^*\otimes 42^*$.
\item $105\mid S^2(21^*)$.
\item $114$ lies inside $24\otimes 27$.
\item $154$ lies inside $\Lambda^2(28^*)$.
\item $162$ is a composition factor of $21^*\otimes 15_2^*$.
\item $210$ is a submodule of $15_1\otimes 28^*$.
\item $273$ is a submodule of $21\otimes 28$.
\end{enumerate}

\subsection*{Extension information}

\begin{center}\begin{tabular}{c|c}
$M$&$\Ext^1=k$
\\\hline $1$& $36$, $36^*$, $215$.
\\$3$&$33^*$, $37$, $71^*$, $75^*$, $162$.
\\$6$&$36^*$, $71^*$, $75^*$, $114$, $117$.
\\$8$&$36$, $36^*$, $75$, $75^*$, $114$, $114^*$, $117$.
\\$10$&$33$, $71$, $75$, $114^*$, $162^*$.
\\$15_1$&$28^*$, $33$, $71$, $162^*$, $210^*$.
\\$15_2$&$37$, $71$, $75$, $75^*$, $114^*$, $117$, $162^*$, 
\\$21$&$21^*$, $63^*$, $215$.
\\$24$&$36$, $36^*$, $63^*$, $114^*$, $117$, $154$, $215$.
\\$27$&$37$, $75$, $75^*$, $105$, $105^*$, $162$, $162^*$.
\\$28$&$15_1^*$, $273$.
\\$33$&$3^*$, $10$, $15_1$, $35^*$.
\\$35$&$33^*$, $105^*$, $162$, $273$.
\\$36$&$1$, $6^*$, $8$, $24$, $24^*$.
\\$37$&$3$, $3^*$, $15_2$, $15_2^*$, $27$.
\\$42$& $71$, $114^*$, $154$, $210^*$.
\\$63$& $21^*$, $24^*$.
\\$71$& $3^*$, $6^*$, $10$, $15_1$, $15_2$, $42$.
\\$75$& $3^*$, $6^*$, $8$, $10$, $15_2$, $15_2^*$, $27$.
\\$105$& $27$, $35^*$.
\\$114$& $6$, $8$, $10^*$, $15_2^*$, $24^*$, $42^*$.
\\$117$& $6$, $6^*$, $8$, $15_2$, $15_2^*$, $24$, $24^*$.
\\$154$& $24$, $42$.
\\$162$& $3$, $10^*$, $15_1^*$, $15_2^*$, $27$, $35$.
\\$210$& $15_1^*$, $42^*$.
\\$215$& $1$, $21$, $21^*$, $24$, $24^*$.
\\$273$& $28$, $35$.
\end{tabular}\end{center}



\subsection*{Tensor products with the Steinberg}

\begin{center}
\begin{tabular}{c|c}
$M$&$M\otimes 343$
\\ \hline $3$&$P(273)$
\\$6$&$P(210)\oplus P(273)$
\\$8$&$P(215)\oplus 343^{\oplus 2}$
\\$10$&$P(154)\oplus P(215)\oplus 343$
\\$15_1$&$P(105)\oplus P(162)\oplus P(210)\oplus P(273)$
\\$15_2$&$P(162)\oplus P(210)\oplus P(273)^{\oplus 2}$
\\$21$&$P(63)\oplus P(114)\oplus P(162)\oplus P(210)\oplus P(273)$
\\$24$&$P(114)\oplus P(162)\oplus P(210)^{\oplus 2}\oplus P(273)^{\oplus 2}$
\\$27$&$P(117)\oplus P(154)\oplus P(154)^*\oplus P(215)^{\oplus 2}\oplus 343^{\oplus 3}$
\\$28$&$P(28)\oplus P(71)\oplus P(117)\oplus P(154)\oplus P(154^*)\oplus P(215)\oplus 343$
\\$33$&$P(33)\oplus P(75)\oplus P(105)\oplus P(114)\oplus P(162)\oplus P(210)\oplus P(273)$
\\$35$&$P(71^*)\oplus P(117)\oplus P(154)\oplus P(154^*)^{\oplus 2}\oplus P(215)^{\oplus 2}\oplus 343^{\oplus 2}$

\end{tabular}\end{center}


\newpage
\section{\texorpdfstring{$\SL_4(2)$}{SL(4,2)}}

\subsection*{Block information}

\begin{center}\begin{tabular}{ccc}
\hline Block & Kernel & Modules
\\\hline Block 1 & $1$ & $1$, $4$, $4^*$, $6$, $14$, $20$, $20^*$
\\Block 2 & $1$ & $64$
\\\hline\end{tabular}\end{center}

\subsection*{Identification of modules}
\begin{enumerate}
\item Fix $4$.
\item $20$ is a submodule of $4\otimes 6$.
\end{enumerate}

\subsection*{Highest weight labels}

\begin{center}\begin{tabular}{ccccccccc}
\hline $\lambda$&000&001&100&010&101&011&110&111
\\\hline $L(\lambda)$ &$1$&$4$&$4^*$&$6$&$14$&$20$&$20^*$&$64$
\\ $\dim(W(\lambda))$&$1$&$4$&$4$&$6$&$15$&$20$&$20$&$64$
\\\hline
\end{tabular}\end{center}

\subsection*{Extension information}

\begin{center}\begin{tabular}{c|ccccccc}
Block 1&$1$&$4$&$4^*$&$6$&$14$&$20$&$20^*$
\\\hline $1$&$0$&$0$&$0$&$1$&$1$&$1$&$1$
\\ $4$&$0$&$0$&$1$&$1$&$0$&$1$&$0$
\\ $4$&$0$&$1$&$0$&$1$&$0$&$0$&$1$
\\ $6$&$1$&$1$&$1$&$0$&$1$&$0$&$0$
\\ $14$&$1$&$0$&$0$&$1$&$0$&$0$&$0$
\\ $20$&$1$&$1$&$0$&$0$&$0$&$0$&$0$
\\ $20^*$&$1$&$0$&$1$&$0$&$0$&$0$&$0$
\end{tabular}\end{center}

\newpage
\section{\texorpdfstring{$\SL_4(3)$}{SL(4,3)}}

\subsection*{Block information}

\begin{center}\begin{tabular}{ccc}
\hline Block & Kernel & Modules
\\\hline Block 1 & $2$ & $1$, $6$, $10$, $10^*$, $15$, $19$, $44$, $45$, $45^*$, $69$, $126$, $126^*$, $156$, $294$
\\Block 2 & $2$ & $729$
\\ Block 3& $1$ & $4$, $4^*$, $16$, $16^*$, $36$, $36^*$, $60$, $60^*$, $116$, $116^*$, $360$, $360^*$
\\\hline\end{tabular}\end{center}

\subsection*{Identification of modules}
\begin{enumerate}
\item Fix $4$.
\item $S^2(4)=10$.
\item $S^3(4)=16/4$.
\item $4\otimes 15=4/16^*/4\oplus 36$.
\item $4\otimes 19=16^*\oplus 60^*$.
\item $16\otimes 10^*=36\oplus 4/116^*/4$.
\item $4\otimes 729=P(360)$.
\item $\Lambda^2(10)=45^*$.
\item $126^*\mid S^3(10)$.
\end{enumerate}

\subsection*{Highest weight labels}

\begin{center}\begin{tabular}{cccccccccccccccc}
\hline $\lambda$&000&010&200&002&101&020&111&012&210&202&022&220&121&212&222
\\\hline $L(\lambda)$ &$1$&$6$&$10$&$10^*$&$15$&$19$&$44$&$45$&$45^*$&$69$&$126$&$126^*$&$156$&$294$&$729$
\\ $\dim(W(\lambda))$&$1$&$6$&$10$&$10$&$15$&$20$&$64$&$45$&$45$&$84$&$126$&$126$&$175$&$300$&$729$
\\\hline
\end{tabular}\end{center}

\begin{center}\begin{tabular}{ccccccccccccc}
\hline $\lambda$&100&001&110&011&201&102&021&120&211&112&221&122
\\\hline $L(\lambda)$ &$4$&$4^*$&$16$&$16^*$&$36$&$36^*$&$60$&$60^*$&$116$&$116^*$&$360$&$360^*$
\\ $\dim(W(\lambda))$&$4$&$4$&$20$&$20$&$36$&$36$&$60$&$60$&$140$&$140$&$360$&$360$
\\\hline
\end{tabular}\end{center}

\subsection*{Extension information}

\begin{center}\begin{tabular}{c|cccccccccccccc}
Block 1&$1$&$6$&$10$&$10^*$&$15$&$19$&$44$&$45$&$45^*$&$69$&$126$&$126^*$&$156$&$294$
\\\hline $1$&$0$&$0$&$0$&$0$&$0$&$1$&$2$&$0$&$0$&$1$&$0$&$0$&$0$&$0$
\\$6$&$0$&$0$&$0$&$0$&$0$&$2$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$2$
\\$10$&$0$&$0$&$0$&$0$&$0$&$1$&$1$&$1$&$0$&$0$&$0$&$0$&$1$&$1$
\\$10^*$&$0$&$0$&$0$&$0$&$0$&$1$&$1$&$0$&$1$&$0$&$0$&$0$&$1$&$1$
\\$15$&$0$&$0$&$0$&$0$&$0$&$0$&$2$&$0$&$0$&$2$&$1$&$1$&$0$&$0$
\\$19$&$1$&$2$&$1$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$1$&$0$
\\$44$&$2$&$1$&$1$&$1$&$2$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\$45$&$0$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$1$&$0$&$0$&$0$
\\$45^*$&$0$&$0$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$1$&$0$&$0$
\\$69$&$1$&$0$&$0$&$0$&$2$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\$126$&$0$&$0$&$0$&$0$&$1$&$0$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$0$
\\$126^*$&$0$&$0$&$0$&$0$&$1$&$0$&$0$&$0$&$1$&$0$&$0$&$0$&$0$&$0$
\\$156$&$0$&$0$&$1$&$1$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\$294$&$0$&$2$&$1$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\end{tabular}\end{center}

\begin{center}\begin{tabular}{c|cccccccccccc}
Block 3&$4$&$4^*$&$16$&$16^*$&$36$&$36^*$&$60$&$60^*$&$116$&$116^*$&$360$&$360^*$
\\ \hline $4$&$0$&$0$&$1$&$1$&$0$&$0$&$2$&$0$&$2$&$2$&$0$&$0$
\\ $4^*$&$0$&$0$&$1$&$1$&$0$&$0$&$0$&$2$&$2$&$2$&$0$&$0$
\\ $16$&$1$&$1$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$1$&$0$
\\ $16^*$&$1$&$1$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$1$
\\ $36$&$0$&$0$&$0$&$0$&$0$&$1$&$1$&$0$&$1$&$0$&$0$&$0$
\\ $36^*$&$0$&$0$&$0$&$0$&$1$&$0$&$0$&$1$&$0$&$1$&$0$&$0$
\\ $60$&$2$&$0$&$0$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\ $60^*$&$0$&$2$&$0$&$0$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$0$
\\ $116$&$2$&$2$&$0$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\ $116^*$&$2$&$2$&$0$&$0$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$0$
\\ $360$&$0$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\ $360^*$&$0$&$0$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\end{tabular}\end{center}


\newpage
\section{\texorpdfstring{$\SL_4(5)$}{SL(4,5)}}

\subsection*{Block information}

\begin{center}\begin{tabular}{ccc}
\hline Block & Kernel & Modules
\\\hline Block 1 & 4 & $1$, $15$, $20_1$, $35$, $35^*$, $45$, $45^*$, $83$, $85$, $173$, $173^*$, $175$, $235$, $235^*$, $299$, $300_1$, $360$, $360^*$, $525$,
\\ && $720_1$, $875$, $875^*$, $885$, $885^*$, $1375$, $1375^*$, $1396$, $1920$, $3420$, $3420^*$, $5025$, $6935$
\\Block 2 & 4 & $15625$
\\ Block 3& 2 & $6$, $10$, $10^*$, $50$, $58$, $68$, $68^*$, $70$, $70^*$, $140_1$, $140_1^*$, $184$, $270$, $270^*$, $300_2$, $420_1$, $530$, $530^*$,
\\ && $560$, $560^*$, $750$, $750^*$, $1190$, $1270$, $1270^*$, $1780$, $1780^*$, $2150$, $3686$, $6500$, $6500^*$, $9400$
\\ Block 4 & 1 &  $4$, $20_2$, $20_3$, $32$, $52$, $60$, $80$, $120$, $140_2$, $160$, $192$, $220$, $260$, $380$, $420_2$, $460$, $476$, $500$, $720_2$,
\\ && $900$, $960$, $1000$, $1400$, $1420$, $1684$, $2940$, $3500$, $4060$, $6060$, $10500$
\\ Block 5 & 1 & Dual to Block 4
\\\hline\end{tabular}\end{center}

\subsection*{Identification of modules}

Fix $4$.

\begin{enumerate}
\item $S^2(4)=10$.
\item $S^2(20_2)$ has $68$ as a submodule.
\item $\Lambda^2(20_3)$ has $70$ as a summand.
\item $\Lambda^2(20_2)$ has $140_1$ as a summand.
\end{enumerate}

\subsection*{Extension information}

\begin{center}
\begin{tabular}{ccccc}\hline
Module & Tested to & $\Ext^1=k$ & $\Ext^1=k^2$
\\ \hline $1$ & All & $83$, $173$, $173^*$ & $299$, $1396$
\\ $15$ & $360$ & $173$, $173^*$, $235$, $235^*$, $299$ & None
\\ $20_1$ & $300$& $85$, $235$, $235^*$, $299$ & None
\\ $35$ & $175$ & $85$ & None
\\ $45$ & $175$ & $85$ & None
\\ $83$ & $85$ & $1$ & None
\\ $85$ & $85$ & $20_1$, $35$, $35^*$, $45$, $45^*$ & None
\\ \hline\end{tabular}
\end{center}

\begin{center}
\begin{tabular}{cccc}\hline
Module & Tested to & $\Ext^1=k$ & $\Ext^1=k^2$
\\ \hline $6$ & $184$ & $58$, $68$, $68^*$ & $184$
\\ $10$ & $184$ & $68^*$ & None
\\ $50$ & $184$ & None & None
\\ $58$ & $140$ & $6$, $68$, $68^*$  & None
\\ $68$ & $70$ & $6$, $10^*$, $58$, $68^*$ & None
\\ $70$ & $70$ & None & None 
\\ \hline\end{tabular}
\end{center}

\begin{center}
\begin{tabular}{cccc}\hline
Module & Tested to & $\Ext^1=k$ & $\Ext^1=k^2$
\\ \hline $4$ & $1684$ & $32$, $52$, $192$, $220$, $380$, $720_2$, $960$ & $476$, $1684$
\\ $20_2$ & $160$ & $80$ & None
\\ $20_3$ & $160$ & $80$ & None
\\ $32$ & $140$ & $4$, $52$ & None
\\ $52$ & $140$ & $4$, $32$, $80$ & None
\\ \hline\end{tabular}
\end{center}

\newpage
\section{\texorpdfstring{$\SL_4(7)$}{SL(4,7)}}


\subsection*{Block information}

These modules are only necessarily complete up to dimension 100.

\begin{center}\begin{tabular}{ccc}
\hline Block & Kernel & Modules
\\\hline1 & 2 & $1$, $6$, $10$, $10^*$, $15$, $20_1$, $35$, $35^*$, $45$, $45^*$, $50$, $64$, $70$, $70^*$, $84_1$, $84_2$, $84_2^*$, $105$, $126$, $126^*$,
\\&& $140_1$, $140_1^*$, $149$, $149^*$, $175$, $189$, $189^*$, $196$, $206$, $206^*$, $211$, $211^*$, $231$, $236$, $260$, $260^*$,
\\&& $280_1$, $280_1^*$, $285$, $315$, $315^*$, $334$, $380$, $380^*$, $484$, $514$, $514^*$, $571$, $571^*$, $616$, $616^*$, $630$, $630^*$,
\\&& $665$, $665^*$, $709$, $735$, $769$, $770_1$, $770_1^*$, $770_2$, $770_2^*$, $824$, $875$, $875^*$, $954$, $1026$, $1026^*$, $1084$,
\\&& $1164$, $1164^*$, $1200$, $1200^*$, $1414$, $1414^*$, $1470$, $1480$, $1480^*$, $1540$, $1540^*$, $1680$, $1680^*$,
\\&& $1820$, $1820^*$, \dots
\\2 & 2 & $117649$
\\3 & 1 & $4$, $4^*$, $20_2$, $20_2^*$, $20_3$, $20_3^*$, \dots
\\\hline\end{tabular}\end{center}

\subsection*{Identification of modules}

\begin{enumerate}
\item Fix $4$ and $4^*$.
\item $S^2(4)=10$.
\item $S^4(4)=35$.
\item $149$ is a composition factor of $20_1\otimes 35$.
\end{enumerate}

\subsection*{Extension information}

\begin{center}\begin{tabular}{ccc}\hline
Module & Tested to & $\Ext^1=k$
\\\hline $1$&$571$&$380$, $380^*$
\\$6$&$236$&$149$, $149^*$
\\$10$&$236$&$149$
\\$15$&$236$&None
\\$20_1$&$236$&None
\\$35$&$196$&None
\\$45$&$84$&None
\\$50$&$84$&None
\\$64$&$84$&None
\\$70$&$70$&None
\\$84_1$ & $64$ & None
\\$84_2$ & $64$ & None
\\ \hline\end{tabular}\end{center}


\newpage
\section{\texorpdfstring{$\SU_4(2)$}{SU(4,2)}}

\subsection*{Block information}

\begin{center}\begin{tabular}{ccc}
\hline Block & Kernel & Modules
\\\hline $1$ & 1 & $1$, $4$, $4^*$, $6$, $14$, $20$, $20^*$
\\ $2$ & 1 & $64$
\\\hline\end{tabular}\end{center}

\subsection*{Extension information}

\begin{center}\begin{tabular}{c|ccccccc}
Block 1&$1$&$4$&$4^*$&$6$&$14$&$20$&$20^*$
\\\hline $1$&$0$&$1$&$1$&$0$&$1$&$0$&$0$
\\$4$&$1$&$0$&$0$&$1$&$0$&$0$&$0$
\\$4^*$&$1$&$0$&$0$&$1$&$0$&$0$&$0$
\\$6$&$0$&$1$&$1$&$0$&$1$&$1$&$1$
\\$14$&$1$&$0$&$0$&$1$&$0$&$0$&$0$
\\$20$&$0$&$0$&$0$&$1$&$0$&$0$&$0$
\\$20^*$&$0$&$0$&$0$&$1$&$0$&$0$&$0$
\end{tabular}\end{center}


\subsection*{Actions of unipotent elements on modules}

\begin{center}\begin{tabular}{c|ccccccc}
Block 1&$1$&$4$&$4^*$&$6$&$14$&$20$&$20^*$
\\\hline 2A&$1$&$2,1^2$&$2,1^2$&$2^2,1^2$&$2^6,1^2$&$2^9,1^2$&$2^9,1^2$
\\ 2B&$1$&$2^2$&$2^2$&$2^2,1^2$&$2^6,1^2$&$2^{10}$&$2^{10}$
\\ 4A[A]&$1$&$3,1$&$3,1$&$3^2$&$4^2,3^2$&$4^4,3,1$&$4^4,3,1$
\\ 4B[B]&$1$&$4$&$4$&$4,2$&$4^3,2$&$4^5$&$4^5$
\end{tabular}\end{center}


\newpage
\section{\texorpdfstring{$\SU_4(3)$}{SU(4,3)}}

\subsection*{Block information}

\begin{center}\begin{tabular}{ccc}
\hline Block & Kernel & Modules
\\\hline 1 & $4$ & $1$, $15$, $19$, $45$, $45^*$, $69$, $156$
\\ 2 & $4$ & $729$
\\ 3 & $2$ & $6$, $10$, $10^*$, $44$, $126$, $126^*$, $294$
\\ 4 & $1$ & $4$, $16$, $36$, $60$, $116$, $360$
\\ 5 & $1$ & Dual to block 4
\\\hline\end{tabular}\end{center}

\subsection*{Identification of modules}

\begin{enumerate}
\item $S^2(16)=10\oplus 126$.
\item $\Lambda^2(10)=45$.
\end{enumerate}
\subsection*{Extension information}

\begin{center}\begin{tabular}{c|ccccccc}
Block 1&$1$&$15$&$19$&$45$&$45^*$&$69$&$156$
\\\hline $1$&$0$&$0$&$3$&$0$&$0$&$1$&$0$
\\$15$&$0$&$0$&$2$&$1$&$1$&$2$&$0$
\\$19$&$3$&$2$&$0$&$0$&$0$&$0$&$3$
\\ $45$&$0$&$1$&$0$&$0$&$1$&$0$&$1$
\\ $45^*$&$0$&$1$&$0$&$1$&$0$&$0$&$1$
\\$69$&$1$&$2$&$0$&$0$&$0$&$0$&$0$
\\$156$&$0$&$0$&$3$&$1$&$1$&$0$&$0$
\end{tabular}\end{center}

\begin{center}\begin{tabular}{c|ccccccc}
Block 3&$6$&$10$&$10^*$&$44$&$126$&$126^*$&$294$
\\\hline $6$&$0$&$0$&$0$&$3$&$0$&$0$&$2$
\\ $10$&$0$&$0$&$1$&$2$&$0$&$1$&$1$
\\ $10^*$&$0$&$1$&$0$&$2$&$1$&$0$&$1$
\\ $44$&$3$&$2$&$2$&$0$&$0$&$0$&$0$
\\ $126$&$0$&$0$&$1$&$0$&$0$&$0$&$0$
\\ $126^*$&$0$&$1$&$0$&$0$&$0$&$0$&$0$
\\ $294$&$2$&$1$&$1$&$0$&$0$&$0$&$0$
\end{tabular}\end{center}

\begin{center}\begin{tabular}{c|cccccc}
Block 4&$4$&$16$&$36$&$60$&$116$&$360$
\\\hline $4$&$0$&$3$&$0$&$2$&$3$&$0$
\\ $16$&$3$&$0$&$1$&$0$&$1$&$0$
\\ $36$&$0$&$1$&$0$&$1$&$0$&$1$
\\ $60$&$2$&$0$&$1$&$0$&$0$&$1$
\\ $116$&$3$&$1$&$0$&$0$&$0$&$0$
\\ $360$&$0$&$0$&$1$&$1$&$0$&$0$
\end{tabular}\end{center}

\subsection*{Tensor products with the Steinberg}

\begin{center}\begin{tabular}{c|c}
$M$ & $M\otimes 729$
\\\hline $15$ & $P(156)\oplus 729^{\oplus 3}$
\\ $19$ & $P(69)\oplus P(156)\oplus 729^{\oplus 2}$
\\ \hline $6$ & $P(294)$
\\ $10$ & $P(126^*)\oplus P(294)$
\\ \hline $4$ & $P(360)$
\\ $16$ & $P(116)\oplus P(360)$
\end{tabular}\end{center}


\newpage
\section{\texorpdfstring{$\SU_4(5)$}{SU(4,5)}}

\subsection*{Block information}
\begin{center}\begin{tabular}{ccc}
\hline Block & Kernel & Modules
\\\hline Block 1 & $2$ & $1,6,10,10^*,15,20_1,35,35^*,45,45^*,50,58,68,68^*,70,70^*,83,85,140_1,140_1^*,173,173^*,$
\\ &&$175,184,235,235^*,270,270^*,299,300_1,300_2,360,360^*,420_1,525,530,530^*$,
\\ && $560,560^*,720_1,750,750^*,875,875^*,885,885^*,1190,1270,1270^*,1375,1375^*,1396,$
\\&&$1780,1780^*,1920,2150,3420,3420^*,3686,5025,6500,6500^*,6935,9400$
\\Block 2 & $2$ & $15625$
\\ Block 3 & $1$ &  $4,4^*,20_2,20_2^*,20_3,20_3^*,32,32^*,52,52^*,60,60^*,80,80^*,120,120^*,140_2,140_2^*,160,160^*,$
\\ &&$192,192^*,220,220^*,260,260^*,380,380^*,420_2,420_2^*,460,460^*,476,476^*,500,500^*,$
\\ && $720_2,720_2^*,900,900^*,960,960^*,1000,1000^*,1400,1400^*,1420,1420^*,1684,1684^*,$
\\ &&$2940,2940^*,3500,3500^*,4060,4060^*,6060,6060^*,10500,10500^*$
\\\hline\end{tabular}\end{center}

\subsection*{Extension information}

\begin{center}\begin{tabular}{cccc}\hline
Module & Tested to & $\Ext^1=k$ & $\Ext^1=k^2$
\\\hline $1$ & $300$ & $68,68^*,83$ & $299$
\\ $6$ & $68$ & $58$ & None
\\ $10$ & $68$ & None & None
\\ $15$ & $68$ & $68,68^*$ & None
\\$20_1$ & $68$ & None & None
\\ $35$ & $58$ & None & None
\\ $45$ & $58$ & None & None
\\ $50$ & $58$ & None & None
\\ $58$ & $58$ & $6$ & None
\\ \hline\end{tabular}\end{center}


\newpage
\section{\texorpdfstring{$\SU_4(7)$}{SU(4,7)}}


\subsection*{Block information}
\begin{center}\begin{tabular}{ccc}
\hline Block & Kernel & Modules
\\\hline Block 1 & $4$ & $1,15,20_1,35,35^*,45,45^*,84_1,105,149,149^*,175,189,189^*,211,211^*,231$,
\\&&$280_1,280_1^*,285,315,315^*,571,571^*,616,616^*,665,665^*,709,735,824,875,875^*$,
\\&&$1164,1164^*,1200,1200^*,1480,1480^*,1540,1540^*,2156...$
\\Block 2 & $4$ & $117649$
\\ Block 3& $2$ & $6,10,10^*,50,64,70,70^*,84_2,84_2^*,126,126^*,140,140^*,196,206,206^*,236,260,260^*$,
\\&&$334,380,380^*,514,514^*,630,630^*,770_1,770_1^*,770_2,770_2^*,954...$
\\ Block 4 & $1$ & $4,20_2,20_3,36,56,60,84_3,100,116,124,140_1,140_2,180,224,280_2,280_3,300$,
\\&&$416,420_1,420_2,480,496,520,756,924,...$
\\ Block 5 & $1$ & Dual to Block 4
\\\hline\end{tabular}\end{center}

\subsection*{Extension information}

\begin{center}\begin{tabular}{ccc}
\hline Module & Tested to & $\Ext^1=k$
\\\hline $1$ & $571$ & $149,149^*$
\\$15$ & $84$ & None
\\$20$ & $45$ & None
\\$35$ & $45$ & None
\\$45$ & $45$ & None
\\\hline\end{tabular}\end{center}

\newpage
\section{\texorpdfstring{$\SL_5(2)$}{SL(5,2)}}

\subsection*{Block information}

\begin{center}\begin{tabular}{ccc}
\hline Block & Kernel & Modules
\\\hline Block 1 & $1$ & $1$, $5$, $5^*$, $10$, $10^*$, $24$, $40_1$, $40_1^*$, $40_2$, $40_2^*$, $74$, $160$, $160^*$, $280$, $280^*$
\\Block 2 & $1$ & $1024$
\\\hline\end{tabular}\end{center}

\subsection*{Identification of modules}

Fix $5$ and $5^*$.

\begin{enumerate}
\item $\Lambda^2(5^*)=10$;
\item $\Lambda^2(10)=5\oplus 40_1$;
\item $5\otimes 10^*=10\oplus 40_2$;
\item $160$ is a composition factor of $10\otimes 24$;
\item $280$ is a composition factor of $5\otimes 74$.
\end{enumerate}

\subsection*{Highest weight labels}

\begin{center}\begin{tabular}{ccccccccc}
\hline $\lambda$ &0000&1000&0001&0010&0100&1001&0101&1010
\\\hline $L(\lambda)$&$1$&$5$&$5^*$&$10$&$10^*$&$24$&$40_1$&$40_1^*$
\\ $\dim(W(\lambda))$&$1$&$5$&$5$&$10$&$10$&$24$&$45$&$45$
\\ \hline $\lambda$&1100&0011&0110&1101&1011&1110&0111&1111
\\\hline $L(\lambda)$ &$40_2$&$40_2^*$&$74$&$160$&$160^*$&$280$&$280^*$&$1024$
\\ $\dim(W(\lambda))$&$40$&$40$&$75$&$175$&$175$&$280$&$280$&$1024$
\\ \hline
\end{tabular}\end{center}



\subsection*{Extension information}

\begin{center}\begin{tabular}{c|ccccccccccccccc}
Block 1&$1$&$5$&$5^*$&$10$&$10^*$&$24$&$40_1$&$40_1^*$&$40_2$&$40_2^*$&$74$&$160$&$160^*$&$280$&$280^*$
\\\hline $1$&$0$&$0$&$0$&$0$&$0$&$0$&$1$&$1$&$0$&$0$&$1$&$0$&$0$&$0$&$0$
\\ $5$&$0$&$0$&$0$&$0$&$1$&$0$&$1$&$0$&$0$&$1$&$1$&$1$&$1$&$0$&$0$
\\ $5^*$&$0$&$0$&$0$&$1$&$0$&$0$&$0$&$1$&$1$&$0$&$1$&$1$&$1$&$0$&$0$
\\ $10$&$0$&$0$&$1$&$0$&$1$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$1$
\\ $10^*$&$0$&$1$&$0$&$1$&$0$&$0$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$1$&$0$
\\ $24$&$0$&$0$&$0$&$0$&$0$&$0$&$1$&$1$&$1$&$1$&$0$&$0$&$0$&$0$&$0$
\\ $40_1$&$1$&$1$&$0$&$1$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\ $40_1^*$&$1$&$0$&$1$&$0$&$1$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\ $40_2$&$0$&$0$&$1$&$0$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$1$&$0$
\\ $40_2^*$&$0$&$1$&$0$&$0$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$1$
\\ $74$&$1$&$1$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\ $160$&$0$&$1$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\ $160^*$&$0$&$1$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\ $280$&$0$&$0$&$0$&$0$&$1$&$0$&$0$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$0$
\\ $280^*$&$0$&$0$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$1$&$0$&$0$&$0$&$0$&$0$
\end{tabular}\end{center}

\newpage
\section{\texorpdfstring{$\SL_5(3)$}{SL(5,3)}}

\subsection*{Block information}

\begin{center}\begin{tabular}{ccc}
\hline Block & Kernel & Modules
\\\hline $1$ & $1$ & $1,5,5^*,10,10^*,15,15^*,24,30,30^*,45_1,45_1^*, 45_2,45_2^*,51,65,65^*,105,105^*,$
\\&&$126,126^*,135,135^*,165,165^*,174,174^*,195,195^*,199,315,315^*,375,375^*,390,390^*,$
\\ && $420,420^*,450,450^*,476,510,510^*,720,720^*,835,835^*,855,855^*,1125,1215,1215^*,$
\\&&$1440,1440^*,1545,1545^*,1701,1701^*,2430,2430^*,2535,2535^*,2574,2574^*,3465,3465^*,$
\\&&$3780,3780^*,4199,6810,6810^*,7350,7350^*,8505,8505^*,10824,19260,19260^*,26730,26730^*$
\\ $2$ & $1$ & $59049$
\\\hline\end{tabular}\end{center}


\subsection*{Extension information}

\begin{center}\begin{tabular}{cccc}\hline
Module & Tested to & $\Ext^1=k$ & $\Ext^1=k^2$
\\\hline $1$ & $6810$ & $135,135^*,174,174^*,199,510,510^*,835,835^*,2535,2535^*$ & $4199$
\\ $5$ & $65$ & $30,45_1,65$ & None
\\ $10$ & $51$ & $30$, $45_1$ & None
\\ $15$ & $51$ & $45_1$ & None
\\ $24$ & $51$ & $51$ & None
\\ $30$ & $51$ & $5$, $10$, $51$ & None
\\ $45_1$ & $51$ & $5,10,15,45_1^*$ & None
\\ $45_2$ & $51$ & None & None
\\ $51$ & $51$ & $24,30,30^*$ & None
\\ \hline\end{tabular}\end{center}


\newpage
\section{\texorpdfstring{$\SL_5(5)$}{SL(5,5)}}

\subsection*{Block information}

These modules are only necessarily complete up to dimension 1200.

\begin{center}\begin{tabular}{ccc}
\hline Block & Kernel & Modules
\\\hline $1$ & $1$ & $1,5,5^*,10,10^*,15,15^*,23,35,35^*,40,40^*,45,45^*,50,50^*,70_1,70_1^*,70_2,70_2^*,75,$
\\&&$103,103^*,105,105^*,121,121^*,160,160^*,170,170^*,175_1,175_1^*,175_2,175_2^*,175_3,175_3^*,$
\\&&$185,185^*,200,235,235^*,255,255^*,280,280^*,305,305^*,315,315^*,320,320^*,365,365^*,381,$
\\&&$410,410^*,420,420^*,450_1,450_1^*,450_2,450_2^*,535,535^*,540,540^*,545,545^*,615,615^*,670,670^*,$
\\ &&$683,683^*,700_1,700_1^*,700_2,700_2^*,794,875,875^*,895,895^*,955,955^*,1000,1050_1,1050_1^*$
\\ &&$1050_2,1050_2^*,1055,1055^*,1063,1063^*,1090,1090^*,1115,1115^*,1150,1150^*,1200,1200^*$,...
\\ $2$ & $1$ & $9765625$
\\\hline\end{tabular}\end{center}

\subsection*{Extension information}

\begin{center}\begin{tabular}{ccc}\hline
Module & Tested to & $\Ext^1=k$
\\\hline $1$ & $1150$ & $23,535,535^*,794$
\\ $5$ & $121$ & $121$
\\ $10$ & $50$ & None
\\ $15$ & $50$ & None
\\ $23$ & $50$ & $1$
\\ $35$ & $50$ & None
\\ $40$ & $50$ & None
\\ $45$ & $50$ & None
\\ $50$ & $50$ & None
\\ \hline\end{tabular}\end{center}



\newpage
\section{\texorpdfstring{$\SU_5(2)$}{SU(5,2)}}


\subsection*{Block information}

\begin{center}\begin{tabular}{ccc}
\hline Block & Kernel & Modules
\\\hline Block 1 & $1$ & $1$, $5$, $5^*$, $10$, $10^*$, $24$, $40_1$, $40_1^*$, $40_2$, $40_2^*$, $74$, $160$, $160^*$, $280$, $280^*$
\\Block 2 & $1$ & $1024$
\\\hline\end{tabular}\end{center}

\subsection*{Identification of modules}

Fix $5$ and $5^*$.

\begin{enumerate}
\item $\Lambda^2(5^*)=10$;
\item $\Lambda^2(10)=5\oplus 40_1$;
\item $5\otimes 10^*=10\oplus 40_2$;
\item $160$ is a composition factor of $10\otimes 24$;
\item $280$ is a submodule of $5\otimes 74$.
\end{enumerate}

\subsection*{Extension information}

\begin{center}\begin{tabular}{c|ccccccccccccccc}
Block 1&$1$&$5$&$5^*$&$10$&$10^*$&$24$&$40_1$&$40_1^*$&$40_2$&$40_2^*$&$74$&$160$&$160^*$&$280$&$280^*$
\\\hline $1$&$0$&$0$&$0$&$1$&$1$&$0$&$0$&$0$&$0$&$0$&$1$&$0$&$0$&$0$&$0$
\\$5$&$0$&$0$&$1$&$1$&$0$&$0$&$1$&$0$&$1$&$0$&$0$&$1$&$0$&$0$&$0$
\\$5^*$&$0$&$1$&$0$&$0$&$1$&$0$&$0$&$1$&$0$&$1$&$0$&$0$&$1$&$0$&$0$
\\$10$&$1$&$1$&$0$&$0$&$0$&$1$&$1$&$1$&$0$&$0$&$0$&$0$&$1$&$0$&$0$
\\$10^*$&$1$&$0$&$1$&$0$&$0$&$1$&$1$&$1$&$0$&$0$&$0$&$1$&$0$&$0$&$0$
\\$24$&$0$&$0$&$0$&$1$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\$40_1$&$0$&$1$&$0$&$1$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$1$
\\$40_1^*$&$0$&$0$&$1$&$1$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$1$&$0$
\\$40_2$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$1$&$0$&$1$&$0$&$0$
\\$40_2^*$&$0$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$1$&$1$&$0$&$0$&$0$
\\$74$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$1$&$1$&$0$&$0$&$0$&$0$&$0$
\\$160$&$0$&$1$&$0$&$0$&$1$&$0$&$0$&$0$&$0$&$1$&$0$&$0$&$0$&$0$&$0$
\\$160^*$&$0$&$0$&$1$&$1$&$0$&$0$&$0$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$0$
\\$280$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\$280^*$&$0$&$0$&$0$&$0$&$0$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\end{tabular}\end{center}

\newpage
\section{\texorpdfstring{$\SU_5(3)$}{SU(5,3)}}

\subsection*{Block information}

\begin{center}\begin{tabular}{ccc}
\hline Block & Kernel & Modules
\\\hline $1$ & $1$ & $1,5,5^*,10,10^*,15,15^*,24,30,30^*,45_1,45_1^*, 45_2,45_2^*,51,65,65^*,105,105^*,$
\\&&$126,126^*,135,135^*,165,165^*,174,174^*,195,195^*,199,315,315^*,375,375^*,390,390^*,$
\\ && $420,420^*,450,450^*,476,510,510^*,720,720^*,835,835^*,855,855^*,1125,1215,1215^*,$
\\&&$1440,1440^*,1545,1545^*,1701,1701^*,2430,2430^*,2535,2535^*,2574,2574^*,3465,3465^*,$
\\&&$3780,3780^*,4199,6810,6810^*,7350,7350^*,8505,8505^*,10824,19260,19260^*,26730,26730^*$
\\ $2$ & $1$ & $59049$
\\\hline\end{tabular}\end{center}


\subsection*{Extension information}

\begin{center}\begin{tabular}{ccc}\hline
Module & Tested to & $\Ext^1=k$
\\\hline $1$ & $199$ & $45_1,45_1^*,174,174^*,199$
\\ $5$ & $65$ & $30,45_1,51,65$
\\ $10$ & $65$ & $30$
\\ $15$ & $65$ & $45_1$
\\ $24$ & $65$ & $45_1,45_1^*,51$
\\ $30$ & $65$ & $5,10,30^*$
\\ $45_1$ & $51$ & $5,15,24$
\\ $45_2$ & $51$ & None
\\ $51$ & $51$ & $5,24$
\\ \hline\end{tabular}\end{center}



\newpage
\section{\texorpdfstring{$\SU_5(5)$}{SU(5,5)}}

\subsection*{Block information}

These modules are only necessarily complete up to dimension 1200.

\begin{center}\begin{tabular}{ccc}
\hline Block & Kernel & Modules
\\\hline $1$ & $1$ & $1,5,5^*,10,10^*,15,15^*,23,35,35^*,40,40^*,45,45^*,50,50^*,70_1,70_1^*,70_2,70_2^*,75,$
\\&&$103,103^*,105,105^*,121,121^*,160,160^*,170,170^*,175_1,175_1^*,175_2,175_2^*,175_3,175_3^*,$
\\&&$185,185^*,200,235,235^*,255,255^*,280,280^*,305,305^*,315,315^*,320,320^*,365,365^*,381,$
\\&&$410,410^*,420,420^*,450_1,450_1^*,450_2,450_2^*,535,535^*,540,540^*,545,545^*,615,615^*,670,670^*,$
\\ &&$683,683^*,700_1,700_1^*,700_2,700_2^*,794,875,875^*,895,895^*,955,955^*,1000,1050_1,1050_1^*$
\\ &&$1050_2,1050_2^*,1055,1055^*,1063,1063^*,1090,1090^*1115,1115^*,1150,1150^*,1200,1200^*$,...
\\ $2$ & $1$ & $9765625$
\\\hline\end{tabular}\end{center}

\subsection*{Extension information}

\begin{center}\begin{tabular}{ccc}\hline
Module & Tested to & $\Ext^1=k$
\\\hline $1$ & $235$ & $23,185,185^*$
\\ $5$ & $45$ & None
\\ $10$ & $45$ & None
\\ $15$ & $45$ & None
\\ $23$ & $45$ & $1$
\\ $35$ & $45$ & None
\\ $40$ & $45$ & None
\\ $45$ & $45$ & None
\\ \hline\end{tabular}\end{center}

\newpage
\section{\texorpdfstring{$3\cdot\Sp_4(2)'$}{3.Sp(4,2)'}}


\subsection*{Block information}

\begin{center}\begin{tabular}{ccc}
\hline Block & Kernel & Modules
\\ \hline 1 & 3 & $1$, $4_1$, $4_2$
\\ 2 & 3 & $8_1$
\\ 3 & 3 & $8_2$
\\ 4 & 1 & $3_1$, $3_2^*$, $9$
\\ 5 & 1 & Dual to block 4
\\\hline\end{tabular}\end{center}

\subsection*{Identification of modules} 

\begin{enumerate}
\item Fix $3_1$. Then $3_2$ is the image under the field automorphism.
\item $4_1$ is a submodule of $3_1\otimes 3_2$ (and $4_2$ is a quotient).
\item $8_1$ is a submodule of $3_1\otimes 3_1^*$.
\end{enumerate}

\subsection*{Highest weight labels}

\begin{center}\begin{tabular}{ccccc}
\hline $\lambda$ &00&10&01&11
\\\hline $L(\lambda)$&$1$&$4_1$&$4_2$&$8_1\oplus 8_2$
\\ $\dim(W(\lambda))$&$1$&$4$&$4$&$16$
\\ \hline
\end{tabular}\end{center}


\subsection*{Extension information}

\begin{center}\begin{tabular}{c|ccc}
Block 1&$1$&$4_1$&$4_2$
\\ \hline $1$&$0$&$1$&$1$
\\ $4_1$&$1$&$0$&$0$
\\ $4_2$&$1$&$0$&$0$
\end{tabular}\end{center}

\begin{center}\begin{tabular}{c|ccc}
Block 4&$3_1$&$3_2^*$&$9$
\\ \hline $3_1$&$0$&$1$&$1$
\\ $3_2^*$&$1$&$0$&$1$
\\ $9$&$1$&$1$&$0$
\end{tabular}\end{center}


\newpage
\section{\texorpdfstring{$\Sp_4(3)$}{Sp(4,3)}}

\subsection*{Block information}

\begin{center}\begin{tabular}{ccc}
\hline Block & Kernel & Modules
\\\hline1 & 2 & $1$, $5$, $10$, $14$, $25$
\\2 & 2 & $81$
\\3 & 1 & $4$, $16$, $40$
\\\hline\end{tabular}\end{center}

\subsection*{Highest weight labels}

\begin{center}\begin{tabular}{ccccccc}
\hline $\lambda$&00&01&20&02&21&22
\\\hline $L(\lambda)$ &$1$&$5$&$10$&$14$&$25$&$81$
\\ $\dim(W(\lambda))$&$1$&$5$&$10$&$14$&$35$&$81$
\\\hline
\end{tabular}\end{center}

\begin{center}\begin{tabular}{cccc}
\hline $\lambda$&10&11&12
\\\hline $L(\lambda)$ &$4$&$16$&$40$
\\ $\dim(W(\lambda))$&$4$&$16$&$40$
\\\hline
\end{tabular}\end{center}

\subsection*{Extension information}

\begin{center}\begin{tabular}{c|ccccc}
Block 1&$1$&$5$&$10$&$14$&$25$
\\ \hline $1$&$0$&$1$&$0$&$1$&$0$
\\ $5$&$1$&$1$&$1$&$1$&$1$
\\ $10$&$0$&$1$&$1$&$1$&$1$
\\ $14$&$1$&$1$&$1$&$0$&$0$
\\ $25$&$0$&$1$&$1$&$0$&$0$
\end{tabular}\end{center}

\begin{center}\begin{tabular}{c|ccc}
Block 3&$4$&$16$&$40$
\\ \hline $4$&$1$&$2$&$1$
\\ $16$&$2$&$1$&$1$
\\ $40$&$1$&$1$&$0$
\end{tabular}\end{center}

\subsection*{Socle layers of projective modules}

\[
\begin{array}{c}
1
\\5\oplus14
\\1\oplus1\oplus5\oplus10\oplus14
\\1\oplus5\oplus5\oplus10\oplus14\oplus25
\\1\oplus1\oplus5\oplus10\oplus14
\\5\oplus14
\\1
\end{array}\qquad
\begin{array}{c}
5
\\1\oplus5\oplus10\oplus14\oplus25
\\1\oplus5\oplus5\oplus5\oplus5\oplus10\oplus10\oplus14\oplus25
\\1\oplus1\oplus5\oplus5\oplus5\oplus10\oplus10\oplus10\oplus14\oplus14\oplus25\
\oplus25
\\1\oplus5\oplus5\oplus5\oplus5\oplus10\oplus10\oplus14\oplus25
\\1\oplus5\oplus10\oplus14\oplus25
\\5
\end{array}
\]
\[ \begin{array}{c}
10
\\10\oplus25
\\10\oplus25
\\10\oplus10
\\10\oplus25
\\10\oplus25
\\10\oplus10
\\5\oplus10\oplus14\oplus25
\\1\oplus5\oplus5\oplus10\oplus10\oplus10\oplus14\oplus25
\\1\oplus5\oplus5\oplus5\oplus10\oplus10\oplus10\oplus14\oplus14\oplus25
\\1\oplus5\oplus5\oplus10\oplus10\oplus10\oplus14\oplus25
\\5\oplus10\oplus14\oplus25
\\10
\end{array}\qquad
\begin{array}{c}
14
\\1\oplus5\oplus10
\\1\oplus5\oplus10\oplus14\oplus14\oplus25
\\1\oplus5\oplus5\oplus10\oplus10\oplus14
\\1\oplus5\oplus10\oplus14\oplus14\oplus25
\\1\oplus5\oplus10
\\14
\end{array}\qquad
\begin{array}{c}
25
\\10
\\10
\\25
\\10
\\10
\\25
\\5\oplus10
\\5\oplus10\oplus14\oplus25
\\1\oplus5\oplus5\oplus10\oplus25
\\5\oplus10\oplus14\oplus25
\\5\oplus10
\\25
\end{array}\]

Remark: these socle layers also appear in David Benson's paper \emph{Projective modules for the group of twenty-seven lines on a cubic surface} (p.1019). There are minor errors for $10$ and $25$.

\newpage
\section{\texorpdfstring{$\Sp_4(5)$}{Sp(4,5)}}

\subsection*{Block information}

\begin{center}\begin{tabular}{ccc}
\hline Block & Kernel & Modules
\\\hline 1&2&$1$, $5$, $10$, $13$, $30$, $35_1$, $35_2$, $55$, $68$, $86$, $105$, $115$, $260$, $355$
\\ 2&2&$625$
\\ 3&1&$4$, $12$, $20$, $40$, $52$, $80$, $140_1$, $140_2$, $204$, $420$
\\\hline\end{tabular}\end{center}

\subsection*{Highest weight labels}

\begin{center}\begin{tabular}{cccccccccccccccc}
\hline $\lambda$&00&01&20&02&03&21&40&04&22&23&41&42&24&43&44
\\\hline $L(\lambda)$ &$1$&$5$&$10$&$13$&$30$&$35_1$&$35_2$&$55$&$68$&$86$&$105$&$115$&$260$&$355$&$625$
\\ $\dim(W(\lambda))$&$1$&$5$&$10$&$14$&$30$&$35$&$35$&$55$&$81$&$154$&$105$&$220$&$260$&$390$&$625$
\\\hline
\end{tabular}\end{center}

\begin{center}\begin{tabular}{ccccccccccc}
\hline $\lambda$&10&11&30&12&31&13&32&14&33&34
\\\hline $L(\lambda)$ &$4$&$12$&$20$&$40$&$52$&$80$&$140_1$&$140_2$&$204$&$420$
\\ $\dim(W(\lambda))$&$4$&$16$&$20$&$40$&$64$&$80$&$140$&$140$&$256$&$420$
\\\hline
\end{tabular}\end{center}


\subsection*{Extension information}


\begin{center}\begin{tabular}{c|cccccccccccccc}
Block 1&$1$&$5$&$10$&$13$&$30$&$35_1$&$35_2$&$55$&$68$&$86$&$105$&$115$&$260$&$355$
\\\hline  $1$&$0$&$0$&$0$&$1$&$0$&$0$&$0$&$0$&$1$&$0$&$0$&$1$&$0$&$0$
\\ $5$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$1$&$1$&$0$&$0$&$0$&$1$
\\ $10$&$0$&$0$&$0$&$0$&$1$&$0$&$0$&$1$&$1$&$0$&$0$&$1$&$0$&$1$
\\ $13$&$1$&$0$&$0$&$1$&$0$&$1$&$0$&$0$&$1$&$1$&$0$&$0$&$1$&$0$
\\ $30$&$0$&$0$&$1$&$0$&$0$&$1$&$1$&$0$&$0$&$0$&$0$&$0$&$1$&$0$
\\ $35_1$&$0$&$0$&$0$&$1$&$1$&$1$&$0$&$1$&$0$&$1$&$0$&$0$&$1$&$1$
\\ $35_2$&$0$&$0$&$0$&$0$&$1$&$0$&$0$&$1$&$1$&$0$&$0$&$0$&$0$&$1$
\\ $55$&$0$&$0$&$1$&$0$&$0$&$1$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\ $68$&$1$&$1$&$1$&$1$&$0$&$0$&$1$&$0$&$1$&$1$&$1$&$0$&$0$&$0$
\\$86$&$0$&$1$&$0$&$1$&$0$&$1$&$0$&$0$&$1$&$0$&$0$&$0$&$0$&$0$
\\$105$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$1$&$0$&$1$&$1$&$0$&$0$
\\ $115$&$1$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$1$&$0$&$0$&$0$
\\ $260$&$0$&$0$&$0$&$1$&$1$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\ $355$&$0$&$1$&$1$&$0$&$0$&$1$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\end{tabular}\end{center}

\begin{center}\begin{tabular}{c|cccccccccc}
Block 3 &$4$&$12$&$20$&$40$&$52$&$80$&$140_1$&$140_2$&$204$&$420$
\\\hline $4$&$0$&$1$&$0$&$0$&$1$&$1$&$1$&$0$&$1$&$0$
\\$12$&$1$&$1$&$0$&$1$&$1$&$0$&$0$&$1$&$1$&$0$
\\$20$&$0$&$0$&$0$&$1$&$0$&$2$&$0$&$1$&$0$&$0$
\\$40$&$0$&$1$&$1$&$1$&$0$&$0$&$0$&$1$&$1$&$1$
\\$52$&$1$&$1$&$0$&$0$&$1$&$1$&$1$&$0$&$1$&$0$
\\$80$&$1$&$0$&$2$&$0$&$1$&$0$&$0$&$0$&$0$&$1$
\\$140_1$&$1$&$0$&$0$&$0$&$1$&$0$&$1$&$0$&$0$&$0$
\\$140_2$&$0$&$1$&$1$&$1$&$0$&$0$&$0$&$0$&$0$&$0$
\\$204$&$1$&$1$&$0$&$1$&$1$&$0$&$0$&$0$&$0$&$0$
\\$420$&$0$&$0$&$0$&$1$&$0$&$1$&$0$&$0$&$0$&$0$
\end{tabular}\end{center}


\newpage
\section{\texorpdfstring{$\Sp_4(7)$}{Sp(4,7)}}

\subsection*{Block information}

\begin{center}\begin{tabular}{ccc}
\hline Block & Kernel & Modules
\\\hline 1 & 2 & $1$, $5$, $10$, $14$, $25$, $35_1$, $35_2$, $54$, $71$, $84$, $91$, $105$, $140_1$, $149$, $154$, $199$,
\\ && $206$, $231$, $260$, $315$, $365$, $455$, $595$, $786$, $959$, $1330$, $1645$
\\ 2 & 2 & $2401$
\\ 3 & 1 & $4$, $16$, $20$, $24$, $44$, $56$, $76$, $116$, $140_2$, $140_3$, $180$,
\\&&$224$, $300$, $336$, $420$, $460$, $560$, $580$, $924$, $1180$, $1820$
\\\hline\end{tabular}\end{center}

\subsection*{Highest weight labels}

\begin{center}\begin{tabular}{ccccccccccccccc}
\hline $\lambda$&00&01&20&02&03&21&40&04&22&60&05&41&06&42
\\\hline $L(\lambda)$ &$1$&$5$&$10$&$14$&$25$&$35_1$&$35_2$&$54$&$71$&$84$&$91$&$105$&$140_1$&$149$
\\ $\dim(W(\lambda))$&$1$&$5$&$10$&$14$&$30$&$35$&$35$&$55$&$81$&$84$&$91$&$105$&$140$&$220$
\\\hline $\lambda$&23&25&24&61&44&63&43&62&26&45&64&46&65&66
\\\hline $L(\lambda)$ &$154$&$199$& $206$&$231$&$260$&$315$&$365$&$455$&$595$&$786$&$959$&$1330$&$1645$&$2401$
\\ $\dim(W(\lambda))$&$154$&$405$&$260$&$231$&$625$&$770$&$390$&$455$&$595$&$935$&$1190$&$1330$&$1729$&$2401$
\\\hline
\end{tabular}\end{center}

\begin{center}\begin{tabular}{cccccccccccc}
\hline $\lambda$&10&11&30&12&31&50&13&51&32&14&33
\\\hline $L(\lambda)$ & $4$&$16$&$20$&$24$&$44$&$56$&$76$&$116$&$140_2$&$140_3$&$180$
\\ $\dim(W(\lambda))$&$4$&$16$&$20$&$40$&$64$&$56$&$80$&$160$&$140$&$140$&$256$
\\ \hline
 $\lambda$&15&52&16&34&35&53&54&36&55&56
\\ \hline $L(\lambda)$ & $224$&$300$&$336$&$420$&$460$&$560$&$580$&$924$&$1180$&$1820$
\\ $\dim(W(\lambda))$&$224$&$324$&$336$&$420$&$640$&$560$&$880$&$924$&$1296$&$1820$
\\ \hline
\end{tabular}\end{center}

\subsection*{Extension information}

\begin{center}
\begin{tabular}{c|c}
Block 1 &  $\Ext^1=k$
\\ \hline $1$ & $54$, $149$, $260$
\\ $5$&$25$, $149$, $199$, $315$, $365$, $786$
\\ $10$&$71$, $149$, $206$, $260$, $365$, $786$, $959$
\\ $14$&$260$, $365$, $959$
\\ $25$&$5$, $25$, $71$, $260$, $365$, $786$
\\ $35_1$ & $206$, $315$, $365$, $786$, $959$, $1645$
\\ $35_2$ & $91$, $140_1$, $206$, $959$, $1645$
\\ $54$ & $1$, $71$, $105$, $206$, $786$, $1330$
\\ $71$ & $10$, $25$, $54$, $71$, $149$, $154$, $199$, $260$, $595$, $786$
\\ $84$ & $91$, $140_1$, $206$, $1645$
\\ $91$ & $35_2$, $84$, $105$, $1330$
\\ $105$& $54$, $91$, $140_1$, $154$, $199$, $595$, $1645$
\\ $140_1$ & $35_2$, $84$, $105$
\\ $149$ & $1$, $5$, $10$, $71$, $149$, $206$, $365$, $786$
\\ $154$ & $71$, $105$, $154$, $595$, $786$, $1330$
\\ $199$ & $5$, $71$, $105$, $206$
\\ $206$ & $10$, $35_1$, $35_2$, $54$, $84$, $149$, $199$, $231$
\\ $231$ & $206$, $365$, $959$
\\ $260$ & $1$, $10$, $14$, $25$, $71$, $365$
\\ $315$ & $5$, $35_1$, $455$
\\ $365$ & $5$, $10$, $14$, $25$, $35_1$, $149$, $231$, $260$, $365$, $455$
\\ $455$ & $315$, $365$, $455$
\\ $595$ & $71$, $105$, $154$
\\ $786$ & $5$, $10$, $25$, $35_1$, $54$, $71$, $149$, $154$
\\ $959$ & $10$, $14$, $35_1$, $35_2$, $231$
\\ $1330$ & $54$, $91$, $154$
\\ $1645$ & $35_1$, $35_2$, $84$, $105$
\end{tabular}
\end{center}

\begin{center}
\begin{tabular}{c|cc}
Block 3 & $\Ext^1=k$ & $\Ext^1=k^2$
\\ \hline $4$ & $76$, $116$, $180$, $300$, $460$, $580$ &
\\ $16$ & $24$, $180$, $300$, $420$, $460$, $560$, $580$, $1180$ &
\\ $20$ & $44$, $180$, $224$, $420$, $580$, $1180$ &
\\ $24$ & $16$, $24$, $76$, $300$, $460$, $580$ &
\\ $44$ & $20$, $76$, $116$, $140_3$, $336$, $460$ &
\\ $56$ & $140_3$, $336$ & $224$
\\ $76$ & $4$, $24$, $44$, $76$, $140_2$, $180$, $460$, $580$, $924$, $1180$ &
\\ $116$ & $4$, $44$, $180$, $224$, $300$, $420$, $1180$ &
\\ $140_2$ & $76$, $140_2$, $140_3$, $336$, $460$, $924$ &
\\ $140_3$ & $44$, $56$, $140_2$, $924$, $1180$, $1820$ &
\\ $180$ & $4$, $16$, $20$, $76$, $116$, $180$, $300$, $460$ &
\\ $224$ & $20$, $116$, $1820$ & $56$
\\ $300$ & $4$, $16$, $24$, $116$, $180$, $300$, $420$, $560$, $580$ &
\\ $336$ & $44$, $56$, $140_2$ &
\\ $420$ & $16$, $20$, $116$, $300$ &
\\ $460$ & $4$, $16$, $24$, $44$, $76$, $140_2$, $180$ &
\\ $560$ & $16$, $300$, $560$ &
\\ $580$ & $4$, $16$, $20$, $24$, $76$, $300$ &
\\ $924$ & $76$, $140_2$, $140_3$ &
\\ $1180$ & $16$, $20$, $76$, $116$, $140_3$ &
\\ $1820$ & $140_3$, $224$ &
\end{tabular}
\end{center}



\newpage
\section{\texorpdfstring{$\Sp_6(2)$}{Sp(6,2)}}

\subsection*{Block information}

\begin{center}\begin{tabular}{ccc}
\hline Block & Kernel & Modules
\\\hline  1 & $1$ & $1$, $6$, $8$, $14$, $48$, $64$, $112$
\\ 2 & $1$ & $512$
\\\hline\end{tabular}\end{center}

\subsection*{Highest weight labels}

\begin{center}\begin{tabular}{ccccccccc}
\hline $\lambda$&000&100&001&010&101&110&011&111
\\\hline $L(\lambda)$ &$1$&$6$&$8$&$14$&$48$&$64$&$112$&$512$
\\ $\dim(W(\lambda))$&$1$&$6$&$14$&$14$&$70$&$64$&$126$&$512$
\\\hline
\end{tabular}\end{center}


\subsection*{Extension information}

\begin{center}\begin{tabular}{c|ccccccc}
Block 1& $1$&$6$&$8$&$14$&$48$&$64$&$112$
\\ \hline $1$&$0$&$1$&$0$&$0$&$1$&$0$&$0$
\\ $6$&$1$&$0$&$1$&$2$&$0$&$0$&$0$
\\ $8$&$0$&$1$&$0$&$0$&$0$&$0$&$1$
\\ $14$&$0$&$2$&$0$&$0$&$0$&$1$&$0$
\\ $48$&$1$&$0$&$0$&$0$&$0$&$0$&$1$
\\ $64$&$0$&$0$&$0$&$1$&$0$&$1$&$0$
\\ $112$&$0$&$0$&$1$&$0$&$1$&$0$&$0$
\end{tabular}\end{center}

\subsection*{Tensor products with the Steinberg}

\begin{center}\begin{tabular}{c|c}
$M$ & $M\otimes 512$
\\\hline $6$ & $P(112)\oplus 512$
\\ $8$ & $P(64)\oplus 512$
\\ $14$ & $P(48)\oplus 512^{\oplus 3}$
\\ $48$ & $P(14)\oplus P(64)\oplus P(112)\oplus 512^{\oplus 7}$
\\ $64$ & $P(8)\oplus P(48)\oplus P(64)^{\oplus 2}\oplus P(112)^{\oplus 3}\oplus 512^{\oplus 11}$
\\ $112$ & $P(6)\oplus P(48)\oplus P(64)^{\oplus 3}\oplus P(112)^{\oplus 4}\oplus 512^{\oplus 19}$
\end{tabular}\end{center}

\subsection*{Symmetric and exterior squares of modules}

\begin{center}\begin{tabular}{c|ccc}
Block 1&$6$&$8$&$14$
\\\hline $S^2(M)$ &$1,14/6$&$1/6/14/6/1,8$&$1\oplus (14/6/1/48/1/6/14,14)$
\\ $\Lambda^2(M)$&$1\oplus 14$&$1/6/14/6/1$&$1\oplus (14/6/1/48/1/6/14)$
\end{tabular}\end{center}

\subsection*{Actions of unipotent elements on modules}

\begin{center}\begin{tabular}{c|ccccccc}
Block 1 & $1$&$6$&$8$&$14$&$48$&$64$&$112$
\\\hline 2A&$1$&$2,1^4$&$2^4$&$2^4,1^6$&$2^{24}$&$2^{24},1^{16}$&$2^{56}$
\\ 2B&$1$&$2^2,1^2$&$2^2,1^4$&$2^6,1^2$&$2^{20},1^8$&$2^{32}$&$2^{52},1^8$
\\ 2C&$1$&$2^2,1^2$&$2^4$&$2^6,1^2$&$2^{24}$&$2^{32}$&$2^{56}$
\\ 2D&$1$&$2^3$&$2^4$&$2^6,1^2$&$2^{24}$&$2^{32}$&$2^{56}$
\\ 4A[B],4D[B]&$1$&$3^2$&$3^2,1^2$&$4^2,3^2$&$4^8,3^4,1^4$&$4^{16}$&$4^{24},3^4,1^4$
\\ 4B[C],4C[C]&$1$&$4,1^2$&$4^2$&$4^3,2$&$4^{12}$&$4^{16}$&$4^{28}$
\\ 4E[C]&$1$&$4,2$&$4^2$&$4^3,2$&$4^{12}$&$4^{16}$&$4^{28}$
\\ 8A[A], 8B[D]&$1$&$6$&$6,2$&$8,6$&$8^4,6^2,2^2$&$8^8$&$8^{12},6^2,2^2$
\end{tabular}\end{center}



\subsection*{Traces of semisimple elements on modules}

\begin{center}\begin{tabular}{c|cccccccc}
Block 1,2 & $1$&$6$&$8$&$14$&$48$&$64$&$112$&$512$
\\\hline 3A&$1$&$3$&$-4$&$2$&$-12$&$4$&$-8$&$-16$
\\3B&$1$&$-3$&$-1$&$5$&$3$&$-8$&$-5$&$8$
\\3C&$1$&$0$&$2$&$-1$&$0$&$-2$&$-2$&$-4$
\\5A&$1$&$1$&$-2$&$-1$&$-2$&$-1$&$2$&$2$
\\7A&$1$&$-1$&$1$&$0$&$-1$&$1$&$0$&$1$
\\9A[B]&$1$&$0$&$-1$&$-1$&$0$&$1$&$1$&$-1$
\\15A[A,A]&$1$&$-2$&$1$&$2$&$-2$&$-1$&$2$&$-1$
\end{tabular}\end{center}

\newpage
\section{\texorpdfstring{$\Sp_6(3)$}{Sp(6,3)}}

\subsection*{Block information}

\begin{center}\begin{tabular}{ccc}
\hline Block & Kernel & Modules
\\ \hline Block 1 & $2$ & $1$, $13$, $21$, $57$, $84$, $90$, $189$, $358$, $594$, $623$, $903$, $1302$, $2457$, $4655$
\\ Block 2 & $2$ & $19683$
\\ Block 3 & $1$ & $6$, $14$, $50$, $126$, $216$, $286$, $316$, $378$, $1260$, $1610$, $3450$, $8316$
\\\hline\end{tabular}\end{center}

\subsection*{Highest weight labels}


\begin{center}\begin{tabular}{ccccccccc}
\hline $\lambda$&000&010&200&101&002&020&210&111
\\\hline $L(\lambda)$&$1$&$13$&$21$&$57$&$84$&$90$&$189$&$358$
\\ $\dim(W(\lambda))$&$1$&$14$&$21$&$70$&$84$&$90$&$189$&$512$
\\\hline
$\lambda$&012&202&220&121&022&212&222&
\\\hline $L(\lambda)$&$594$&$623$&$903$&$1302$&$2457$&$4655$&$19683$&
\\ $\dim(W(\lambda))$&$594$&$1078$&$924$&$2205$&$2457$&$5720$&$19683$&
\\ \hline
\end{tabular}\end{center}

\begin{center}\begin{tabular}{ccccccc}
\hline $\lambda$&100&001&110&011&201&120
\\\hline $L(\lambda)$&$6$&$14$&$50$&$126$&$216$&$286$
\\ $\dim(W(\lambda))$&$6$&$14$&$64$&$126$&$216$&$350$
\\\hline
$\lambda$&021&102&211&112&221&122
\\\hline $L(\lambda)$&$316$&$378$&$1260$&$1610$&$3450$&$8316$
\\ $\dim(W(\lambda))$&$616$&$378$&$1386$&$2240$&$5460$&$8316$
\\ \hline
\end{tabular}\end{center}

\subsection*{Extension information}

\begin{center}\begin{tabular}{c|cccccccccccccc}
Module & $1$&$13$&$21$&$57$&$84$&$90$&$189$&$358$&$594$&$623$&$903$&$1302$&$2457$&$4655$
\\ \hline $1$&$0$&$1$&$0$&$0$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$1$
\\ $13$&$1$&$1$&$0$&$1$&$0$&$0$&$0$&$2$&$0$&$1$
\\ $21$&$0$&$0$&$0$&$1$&$1$&$1$&$0$&$0$
\\ $57$&$0$&$1$&$1$&$0$&$0$&$1$&$1$
\\ $84$&$0$&$0$&$1$&$0$&$0$&$1$
\\ $90$&$1$&$0$&$1$&$1$&$1$&$1$
\\ $189$&$0$&$0$&$0$&$1$
\\ $358$&$0$&$2$&$0$&
\\ $594$&$0$&$0$&
\\ $623$&$0$&$1$& 
\\ $903$&$0$& 
\\ $1302$&$0$&
\\ $2457$&$0$&
\\ $4655$&$1$&
\end{tabular}\end{center}

\begin{center}\begin{tabular}{c|cccccccccccc}
Module & $6$&$14$&$50$&$126$&$216$&$286$&$316$&$378$&$1260$&$1610$&$3450$&$8316$
\\ \hline $6$&$0$&$0$&$1$&$1$&$0$&$0$&$0$&$0$&$1$&$1$&&$0$
\\ $14$&$0$&$1$&$1$&$0$&$0$&$1$&&&&&&$0$
\\ $50$&$1$&$1$&$0$&$2$&&&&&&&&$0$
\\ $126$&$1$&$0$&$2$&$1$&&&&&&&&$0$
\\ $216$&$0$&$0$&&&&&&&&&&$0$
\\ $286$&$0$&$1$&&&&&&&&&&$0$
\\ $316$&$0$&&&&&&&&&&&$1$
\\ $378$&$0$&&&&&&&&&&&$1$
\\ $1260$&$1$&&&&&&&&&&&$0$
\\ $1610$&$1$&&&&&&&&&&&$0$
\\ $3450$&&&&&&&&&&&&$0$
\\ $8316$&$0$&$0$&$0$&$0$&$0$&$0$&$1$&$1$&$0$&$0$&$0$&$0$
\end{tabular}\end{center}

\subsection*{Actions of unipotent elements on modules}

\begin{small}
\begin{center}\begin{tabular}{c|cccccc}
Block 1&$1$&$13$&$21$&$57$&$84$&$90$
\\\hline 3A[2]&$1$&$2^4,1^5$&$3,2^4,1^{10}$&$3^5,2^{16},1^{10}$&$3^{14},2^{16},1^{10}$&$3^{10},2^{20},1^{20}$
\\ 3C,3D&$1$&$3,2^4,1^2$&$3^3,2^4,1^4$&$3^{14},2^4,1^7$&$3^{23},2^4,1^7$&$3^{22},2^8,1^8$
\\ 3E[2]&$1$&$3^3,1^4$&$3^6,1^3$&$3^{18},1^3$&$3^{27},1^3$&$3^{27},1^9$
\\ 3G&$1$&$3^4,1$&$3^7$&$3^{19}$&$3^{27},1^3$&$3^{30}$
\\ 9A[A][2]&$1$&$5,4^2$&$7,4^2,3,1^3$&$8^2,7,6^2,5^3,3,2^2$&$9^3,8^2,7,6^2,5^3,3,2^2$&$9,8^2,7^3,6^2,5^2,4^2,3^3,2^2,1$
\\9C[A][2],9E[A][2]&$1$&$5^2,3$&$7,5,3^3$&$9,7^2,6^4,3^3,1$&$9^4,7^2,6^4,3^3,1$&$9^3,7,6^4,5^3,3^5,1^2$
\\9G[E][2]&$1$&$9,3,1$&$9^2,3$&$9^6,3$&$9^9,3$&$9^9,3^3$
\end{tabular}\end{center}

\begin{center}\begin{tabular}{c|ccccc}
Block 1&$189$&$358$&$594$&$623$&$903$
\\\hline 3A[2]&$3^{24},2^{36},1^{45}$&$3^{77},2^{44},1^{39}$&$3^{159},2^{36},1^{45}$&$3^{189},2^{20},1^{16}$&$3^{199},2^{100},1^{106}$
\\ 3C,3D&$3^{51},2^{12},1^{12}$&$3^{113},2^8,1^3$&$3^{186},2^{12},1^{12}$&$3^{199},2^8,1^{10}$&$3^{297},2^4,1^4$
\\ 3E[2]&$3^{63}$&$3^{117},1^7$&$3^{198}$&$3^{201},2^8,1^4$&$3^{297},2^6$
\\ 3G&$3^{63}$&$3^{118},1^4$&$3^{198}$&$3^{206},1^5$&$3^{301}$
\\ 9A[A][2]&$9^77^36^65^34^63^91^3$&$9^{24}8^276^{12}54^23^{10}2^2$&$9^{52}7^36^65^34^63^91^3$&$9^{63}6^25^54^43$&$9^{65}7^46^{30}4^23^{34}$
\\9C[A][2],9E[A][2]&$9^8,6^{12},3^{15}$&$9^{25},7^2,6^{12},5^2,3^{12},1$&$9^{53},6^{12},3^{15}$&$9^{63},6^4,5^4,3^4$&$9^{66},7,6^{32},5,3^{35}$
\\9G[E][2]&$9^{21}$&$9^{39},3^2,1$&$9^{66}$&$9^{67},6^2,5,3$&$9^{99},6^2$
\end{tabular}\end{center}

\begin{center}\begin{tabular}{c|ccccc}
Block 3&$6$&$14$&$50$&$126$&$216$
\\\hline 3A[2]&$2,1^4$&$2^5,1^4$&$3^4,2^{11},1^{16}$&$3^{16},2^{29},1^{20}$&$3^{30},2^{45},1^{36}$
\\3C,3D&$2^2,1^2$&$3^2,2^2,1^4$&$3^{10},2^8,1^4$&$3^{32},2^{10},1^{10}$&$3^{60},2^{12},1^{12}$
\\ 3E[2]&$2^3$&$3^2,2^4$&$3^{14},2^4$&$3^{36},2^9$&$3^{72}$
\\ 3G&$3^2$&$3^4,1^2$&$3^{16},1^2$&$3^{42}$&$3^{72}$
\\9A[A][2]&$4,1^2$&$5^2,4$&$8,7^2,6,4^3,3^2,2,1^2$&$9^2,8^4,7^2,6^4,5^4,4,3^2,2^4$&$9^8,8^3,6^9,5^6,4^3,3^6,2^3$
\\9C[A][2],9E[A][2]&$4,2$&$6,4^2$&$8^2,6^3,3^4,2^2$&$9^4,8^2,6^8,4^3,3^4,2$&$9^{10},6^{15},3^{12}$
\\ 9G[E][2]&$6$&$8,6$&$9^4,8,6$&$9^{12},6^3$&$9^{24}$
\end{tabular}\end{center}

\begin{center}\begin{tabular}{c|ccc}
Block 3&$286$&$316$&$378$
\\\hline 3A[2]&$3^{56},2^{39},1^{40}$&$3^{90},2^{15},1^{16}$&$3^{84},2^{45},1^{36}$
\\ 3C,3D&$3^{92},2^2,1^6$&$3^{98},2^8,1^6$&$3^{114},2^{12},1^{12}$
\\ 3E[2]&$3^{92},2^4,1^2$&$3^{102},2^4,1^2$&$3^{126}$
\\ 3G&$3^{94},1^4$&$3^{104},1^4$&$3^{126}$
\\9A[A][2]&$9^{18},7^2,6^{10},5^2,4,3^{12}$&$9^{28},8,7^4,6,4^3,3^2,2,1^2$&$9^{26},8^3,6^9,5^6,4^3,3^6,2^3$
\\9C[A][2],9E[A][2]&$9^{18},8,6^{12},4^2,3^{12}$&$9^{28},8^3,6^4,3^4,2^2$&$9^{28},6^{15},3^{12}$
\\ 9G[E][2]&$9^{30},8,6,2$&$9^{34},6,4$&$9^{42}$
\end{tabular}\end{center}
\end{small}

\newpage
\section{\texorpdfstring{$\Sp_6(5)$}{Sp(6,5)}}


\subsection*{Block information}

These modules are only necessarily complete up to dimension 2000.

\begin{center}\begin{tabular}{ccc}
\hline Block & Kernel & Modules
\\ \hline Block 1 & 2 & $1$, $14_1$, $21$, $63$, $70$, $90$, $126$, $189$, $295$, $309$, $426$, $525$, $573$, $798$, $813$, $938$, $951_1$, $951_2$,
\\ && $1197$, $1506$, $2135$, $2619$, $3513$, $3585$, $3900$, $4143$, $7314$, $9450$, $12375$, \dots
\\ Block 2 & 2 & $1953125$
\\ Block 3 & 1 & $6$, $14_2$, $56$, $62$, $64$, $202$, $246$, $316$, $330$, $350$, $552$, $666$, $924$, $1100$, $1246$, $1288$, $1644$,
\\ && $1686$, $2240$, $3192$, $3220$, $3794$, $3962$, $4858$, $6300$, $6390$, $6874$, $7686$, \dots
\\\hline\end{tabular}\end{center}


\subsection*{Highest weight labels}

\begin{center}\begin{tabular}{cccccccccccc}
\hline $\lambda$&000&010&200&002&101&020&400&210&030&111&220
\\\hline $L(\lambda)$&$1$&$14_1$&$21$&$63$&$70$&$90$&$126$&$189$&$295$&$309$&$426$
\\ $\dim(W(\lambda))$&$1$&$14$&$21$&$84$&$70$&$90$&$126$&$189$&$385$&$512$&$924$
\\\hline
$\lambda$&301&012&410&103&004&202&040&311&022&121&
\\\hline $L(\lambda)$&$525$&$573$&$798$&$813$&$938$&$951_1$&$951_2$&$1197$&$1506$&$2135$&
\\ $\dim(W(\lambda))$&$525$&$594$&$924$&$1386$&$1001$&$1078$&$1274$&$3072$&$2457$&$2205$&
\\ \hline
\end{tabular}\end{center}

\begin{center}\begin{tabular}{cccccccccc}
\hline $\lambda$&100&001&300&011&110&201&310&102&003
\\\hline $L(\lambda)$&$6$&$14_2$&$56$&$62$&$64$&$202$&$246$&$316$&$330$
\\ $\dim(W(\lambda))$&$6$&$14$&$56$&$126$&$64$&$216$&$448$&$378$&$330$
\\\hline
$\lambda$&120&021&130&211&401&031&302&320&013
\\\hline $L(\lambda)$&$350$&$552$&$666$&$924$&$1100$&$1246$&$1288$&$1644$&$1686$
\\ $\dim(W(\lambda))$&$350$&$616$&$1344$&$1386$&$1100$&$2184$&$2464$&$2016$&$2002$
\\ \hline
\end{tabular}\end{center}


\subsection*{Extension information}

\begin{center}\begin{tabular}{ccc}\hline
Module & Tested to & $\Ext^1=k$
\\\hline $1$ & $2619$ & $426$, $951_1$
\\ $14_1$ & $426$ & $309$, $426$
\\ $21$ & $189$ & $63$
\\ $63$ & $70$ & $21$, $63$
\\ \hline\end{tabular}\end{center}

\newpage
\section{\texorpdfstring{$\Sp_6(7)$}{Sp(6,7)}}

\subsection*{Block information}

These modules are only necessarily complete up to dimension 2000.
\begin{center}\begin{tabular}{ccc}
\hline Block & Kernel & Modules
\\ \hline 1 & 2 & 
$1$, $14_1$, $21$, $70$, $84$, $89$, $126_1$, $171$, $189$, $385$, $423$, $462$, $525$, $861$, $875$, $924_1$, $924_2$, $1078$,
\\ && $1251$, $1274$, $1533$, $2079$, $2205$, $2254$, $2388$, $2649$, $2667$, $3003$, $3276$, $3507$, $4830$,
\\ && $5334$, $5922$, $6041$, $6055$, $6087$, $6776$, $8547$, $11914$, $11319$, $12882$, $15092$, and $132$ more
\\ 2 & 2 & $40353607$
\\ $3$ & 1 &
$6$, $14_2$, $56$, $58$, $126_2$, $158$, $172$, $252$, $350$, $378$, $448$, $616$, $786$, $840$, $854$, $942$,
$1344$,
\\ && $1386$, $1544$, $1876$, $2016$, $2086$, $2414$, $3640$, $3696$, $4116$, $4844_1$, $4844_2$, $5950$, $5964$, $6006$,
\\ && $8918$, and $136$ more
\\\hline\end{tabular}\end{center}


\subsection*{Highest weight labels}

\begin{center}\begin{tabular}{cccccccccccc}
\hline $\lambda$&000&010&200&101&002&020&400&012&210&030&111
\\ \hline $L(\lambda)$&$1$&$14_1$&$21$&$70$&$84$&$89$&$126_1$&$171$&$189$&$385$&$423$
\\ $\dim(W(\lambda))$&$1$&$14$&$21$&$70$&$84$&$90$&$126$&$594$&$189$&$385$&$512$
\\ \hline $\lambda$&600&301&103&004&410&220&202&420&040&022&501
\\ \hline $L(\lambda)$&$462$&$525$&$861$&$875$&$924_1$&$924_2$&$1078$&$1251$&$1274$&$1533$&$2079$
\\ $\dim(W(\lambda))$&$462$&$525$&$1386$&$1001$&$924$&$924$&$1078$&$3900$&$1274$&$2457$&$2079$
\\ \hline
\end{tabular}\end{center}

\begin{center}\begin{tabular}{ccccccccccccccc}
\hline $\lambda$&100&001&300&110&011&201&500&120&310&021&510&031&112&401
\\ \hline $L(\lambda)$&$6$&$14_2$&$56$&$58$&$126_2$&$158$&$252$&$350$&$448$&$616$&$786$&$840$&$854$&$942$
\\ $\dim(W(\lambda))$&$6$&$14$&$56$&$64$&$126$&$216$&$252$&$350$&$448$&$616$&$1728$&$2184$&$2240$&$1100$
\\ \hline
\end{tabular}\end{center}

\subsection*{Extension information}

The trivial module only has $1$-cohomology with $89$ and $1251$ for all modules up to $6776$ listed above.

\begin{center}\begin{tabular}{c|ccccccccc}
Block 1&$1$&$14_1$&$21$&$70$&$84$&$89$&$126_1$&$171$&$189$
\\\hline $1$&$0$&$0$&$0$&$0$&$0$&$1$&$0$&$0$&$0
\\$14_1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\$21$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\$70$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\$84$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\$89$&$1$&$0$&$0$&$0$&$0$&$0$&$0$
\\$126_1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\end{tabular}\end{center}

\begin{center}\begin{tabular}{c|ccccc}
Block 3&$6$&$14_2$&$56$&$58$&$126_2$
\\\hline $6$&$0$&$0$&$0$&$1$&$0$
\\$14_2$&$0$&$0$&$0$&$0$&$0$	
\\$56$&$0$&$0$&$0$&$0$&$0$
\\$58$&$1$&$0$&$0$&$0$&$0$
\\$126$&$0$&$0$&$0$&$0$&$0$
\end{tabular}\end{center}

\newpage
\section{\texorpdfstring{$\Sp_8(2)$}{Sp(8,2)}}

\subsection*{Block information}

\begin{center}\begin{tabular}{ccc}
\hline Block & Kernel & Modules
\\\hline $1$ & $1$ & $1,8,16,26,48,128,160,246,416,768,784,2560,3936,4096,12544$
\\ $2$ & $1$ & $65536$
\\\hline\end{tabular}\end{center}

\subsection*{Highest weight labels}

\begin{center}\begin{tabular}{ccccccccc}
\hline $\lambda$&0000&1000&0001&0100&0010&1001&1100&1010
\\\hline $L(\lambda)$ & $1$&$8$&$16$&$26$&$48$&$128$&$160$&$246$
\\ $\dim(W(\lambda))$&$1$&8&42&27&48&288&160&315
\\\hline $\lambda$&0101&0011&0110&1101&1011&1110&0111&1111
\\\hline $L(\lambda)$&$416$&$768$&$784$&$2560$&$3936$&$4096$&$12544$&$65536$
\\ $\dim(W(\lambda))$&792&1056&792&4200&6237&4096&13728&65536
\\ \hline
\end{tabular}\end{center}


\subsection*{Extension information}

\begin{center}\begin{tabular}{c|cccccccccccccc}
Block 1 &$1$&$8$&$16$&$26$&$48$&$128$&$160$&$246$&$416$&$768$&$784$&$2560$&$3936$&$4096$
\\ \hline 
$1$&$0$&$1$&$0$&$1$&$1$&$0$&$0$&$1$&$1$&$0$&$0$&$0$&$0$&$0$
\\$8$&$1$&$0$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$1$&&&
\\$16$&$0$&$0$&$0$&$1$&$0$&$0$&$0$&$0$&$2$&$1$&&&&
\\$26$&$1$&$1$&$1$&$0$&$1$&$0$&$1$&$2$&&&&&&
\\$48$&$1$&$0$&$0$&$1$&$0$&$0$&$0$&&&&&&&
\\$128$&$0$&$0$&$0$&$0$&$0$&&&&&&&&&
\\$160$&$0$&$0$&$0$&$1$&$0$&&&&&&&&&
\\$246$&$1$&$0$&$0$&$2$&&&&&&&&&&
\\$416$&$1$&$0$&$2$&&&&&&&&&&&
\\$768$&$0$&$0$&$1$&&&&&&&&&&&
\\$784$&$0$&$1$&&&&&&&&&&&&
\end{tabular}\end{center}



\newpage
\section{\texorpdfstring{$\Sp_8(3)$}{Sp(8,3)}}


\subsection*{Block information}

These modules are only necessarily complete up to dimension 4000.

\begin{center}\begin{tabular}{ccc}
\hline Block & Kernel & Modules
\\\hline $1$ & $2$ & $1,27,36,41,266,279,513,594_1,594_2,784,1114,1891,4320,6237,6426,10150$, and 28 more
\\ $2$ & $2$ & $43046721$
\\ $3$ & $1$ & $8,40,112,288,504,1016,1072,1512,2072,2640,4752,6048,9080,$ and 23 more
\\\hline\end{tabular}\end{center}


\subsection*{Highest weight labels}

These are only up to dimension 5000.


\begin{center}\begin{tabular}{cccccccccccccc}
\hline $\lambda$&0000&0100&2000&0001&0200&1010&0101&2100&0002&0020&2001&1110&2200
\\\hline $L(\lambda)$ & $1$&$27$&$36$&$41$&$266$&$279$&$513$&$594_1$&$594_2$&$784$&$1114$&$1891$&$4320$
\\ $\dim(W(\lambda))$&$1$&$27$&$36$&$42$&$308$&$315$&$792$&$594$&$594$&$825$&$1155$&$4096$&$4914$
\\\hline
\end{tabular}\end{center}

\begin{center}\begin{tabular}{cccccccccccc}
\hline $\lambda$&1000&0010&1100&1001&0110&0011&2010&1200&1101&1002&102
\\\hline $L(\lambda)$ & $8$&$40$&$112$&$288$&$504$&$1016$&$1072$&$1512$&$2072$&$2640$&$4752
\\ $\dim(W(\lambda))$&$8$&$48$&$160$&$288$&$792$&$1056$&$1232$&$1512$&$4200$&$3696$&$4752$
\\\hline \end{tabular}\end{center}

\subsection*{Extension information}

\begin{center}\begin{tabular}{ccc}\hline
Module & Tested to & $\Ext^1=k$
\\\hline $1$ & $1891$ & $41$, $266$
\\ $27$ & $41$ & None
\\ $36$ & $41$ & None
\\ $41$ & $41$ & $1,41$
\\ \hline\end{tabular}\end{center}

\begin{center}\begin{tabular}{ccc}\hline
Module & Tested to & $\Ext^1=k$
\\\hline $8$ & $112$ & $40,112$
\\ $40$ & $112$ & $8,40$
\\ \hline\end{tabular}\end{center}

\newpage
\section{\texorpdfstring{$\Sp_8(5)$}{Sp(8,5)}}



\subsection*{Block information}

These modules are only necessarily complete up to dimension 4000.

\begin{center}\begin{tabular}{ccc}
\hline Block & Kernel & Modules
\\\hline $1$ & $2$ & $1,27,36,42,281,313,315,330,558,792,825,1155,1652,2183_1,2183_2,3138,3262$...
\\ $2$ & $2$ & $152587890625$
\\ $3$ & $1$ & $8,48,120,160,288,312,744,784,944,1504,2192,3144,3520$...
\\\hline\end{tabular}\end{center}


\subsection*{Highest weight labels}

These are only up to dimension 4000.


\begin{center}\begin{tabular}{cccccccccc}
\hline $\lambda$&0000&0100&2000&0001&0200&0002&1010&4000&2100
\\\hline $L(\lambda)$ & $1$&$27$&$36$&$42$&$281$&$313$&$315$&$330$&$558$
\\ $\dim(W(\lambda))$&$1$&$27$&$36$&$42$&$308$&$594$&$315$&$330$&$594$
\\\hline $\lambda$&0101&0020&2001&2200&0300&1011&3010&1110&
\\\hline $L(\lambda)$ &$792$&$825$&$1155$& $1652$&$2183_1$&$2183_2$&$3138$&$3262$&
\\ $\dim(W(\lambda))$&$792$&$825$&$1155$&$4914$&$2184$&$6237$&$3696$&$4096$&
\\\hline
\end{tabular}\end{center}

\begin{center}\begin{tabular}{cccccccccccccc}
\hline $\lambda$&1000&0010&3000&1100&1001&0011&0110&3100&2010&1200&1002&1300&3001
\\\hline $L(\lambda)$ & $8$&$48$&$120$&$160$&$288$&$312$&$744$&$784$&$944$&$1504$&$2192$&$3144$&$3520$
\\ $\dim(W(\lambda))$&$8$&$48$&$120$&$160$&$288$&$1056$&$792$&$1728$&$1232$&$1512$&$3696$&$9408$&$3520$
\\\hline
\end{tabular}\end{center}


\subsection*{Extension information}

\begin{center}\begin{tabular}{ccc}\hline
Module & Tested to & $\Ext^1=k$
\\\hline $1$ & $1155$ & None
\\ $27$ & $281$ & $281$
\\ $36$ & $42$ & None
\\ $42$ & $42$ & None
\\ \hline\end{tabular}\end{center}

\begin{center}\begin{tabular}{ccc}\hline
Module & Tested to & $\Ext^1=k$
\\\hline $8$ & $288$ & None
\\ $48$ & $48$ & None
\\ \hline\end{tabular}\end{center}

\newpage
\section{\texorpdfstring{$\Omega_7(3)$}{O7(3)}}

\subsection*{Block information}

\begin{center}\begin{tabular}{ccc}
\hline Block & Kernel & Modules
\\\hline Block 1& $2$ & $1$, $7$, $21$, $27$, $35$, $63$, $132$, $189_1$, $189_2$, $309$, $518$, $672_1$, $797$, $1602$, $2052$, $2855$, $5670$
\\ Block 2 & $2$ & $19683$
\\ Block 3 & $1$ & $8$, $48$, $104$, $120$, $384$, $672_2$, $1512$, $2400$, $7272$
\\\hline\end{tabular}\end{center}


\subsection*{Highest weight labels}


\begin{center}\begin{tabular}{cccccccccc}
\hline $\lambda$&000&100&010&200&002&110&020&102&012
\\\hline $L(\lambda)$ &$1$&$7$&$21$&$27$&$35$&$63$&$132$&$189_1$&$189_2$
\\ $\dim(W(\lambda))$&$1$&$7$&$21$&$27$&$35$&$105$&$168$&$189$&$378$
\\\hline $\lambda$&210&202&120&112&220&022&212&122&222
\\\hline $L(\lambda)$&$309$&$518$&$672_1$&$797$&$1602$&$2052$&$2855$&$5670$&$19683$
\\ $\dim(W(\lambda))$&$330$&$616$&$693$&$1617$&$1911$&$2079$&$4550$&$7722$&$19683$
\\\hline
\end{tabular}\end{center}

\begin{center}\begin{tabular}{cccccccccc}
\hline $\lambda$&001&101&011&201&111&021&211&121&221
\\\hline $L(\lambda)$ &$8$&$48$&$104$&$120$&$384$&$672_2$&$1512$&$2400$&$7272$
\\ $\dim(W(\lambda))$&$8$&$48$&$112$&$168$&$512$&$720$&$1512$&$2800$&$7392$
\\\hline
\end{tabular}\end{center}

\subsection*{Extension information}

\begin{small}\begin{center}\begin{tabular}{c|ccccccccccccccccc}
Block 1 & $1$&$7$&$21$&$27$&$35$&$63$&$132$&$189_1$&$189_2$&$309$&$518$&$672_1$&$797$&$1602$
&$2052$&$2855$&$5670$
\\ \hline $1$&$0$&$0$&$0$&$0$&$0$&$1$&$3$&$0$&$0$&$0$&$0$&$0$&$2$&$0$&$0$&$2$&$0$
\\ $7$&$0$&$0$&$0$&$0$&$0$&$3$&$1$&$0$&$1$&$1$&$0$&&&&&&$0$
\\ $21$&$0$&$0$&$0$&$0$&$0$&$1$&$2$&$0$&$0$&&&&&&&&$0$
\\ $27$&$0$&$0$&$0$&$0$&$0$&$0$&$1$&$0$&$0$&&&&&&&&$0$
\\ $35$&$0$&$0$&$0$&$0$&$0$&$2$&$2$&$0$&$0$&&&&&&&&$0$
\\ $63$&$1$&$3$&$1$&$0$&$2$&$0$&$0$&$0$&$0$&&&&&&&&$0$
\\ $132$&$3$&$1$&$2$&$1$&$2$&$0$&$0$&&&&&&&&&&$1$
\\ $189_1$&$0$&$0$&$0$&$0$&$0$&$0$&&&&&&&&&&&$0$
\\ $189_2$&$0$&$1$&$0$&$0$&$0$&$0$&&&&&&&&&&&$0$
\\ $309$&$0$&$1$&&&&&&&&&&&&&&&$0$
\\ $518$&$0$&$0$&&&&&&&&&&&&&&&$0$
\\ $672_1$&$0$&&&&&&&&&&&&&&&&$0$
\\ $797$&$2$&&&&&&&&&&&&&&&&$0$
\\ $1602$&$0$&&&&&&&&&&&&&&&&$0$
\\ $2052$&$0$&&&&&&&&&&&&&&&&$1$
\\ $2855$&$2$&&&&&&&&&&&&&&&&$0$
\\ $5670$&$0$&$0$&$0$&$0$&$0$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$1$&$0$&$0$
\end{tabular}\end{center}\end{small}


\begin{center}\begin{tabular}{c|cccccc}
Block 3&$8$&$48$&$104$&$120$&$384$&$672_2
\\ \hline $8$&$0$&$0$&$2$&$1$&$6$&$2
\\ $48$&$0$&$0$&$0$&$2$&
\\ $104$&$2$&$0$&$0$&$0$&
\\ $120$&$1$&$2$&$0$&&
\\ $384$&$6$&&&&
\\ $672_2$&$2$&&&&
\end{tabular}\end{center}


\newpage
\section{\texorpdfstring{$\Omega_7(5)$}{O7(5)}}


\subsection*{Block information}

These modules are only necessarily complete up to dimension 2000.

\begin{center}\begin{tabular}{ccc}
\hline Block & Kernel & Modules
\\\hline Block 1& $2$ & $1$, $7$, $21$, $27$, $35$, $77$, $105$, $141$, $168_1$, $182$, $293$, $330$, $371_1$, $371_2$, $448_1$, $483$, $616$, $825$, $988$,
\\ && $1093$, $1246$, $1322$, $1513$, $1560$, $1750$, $2008$, $2310$, $2380$, $2393$, $3375$, $4095$,
$4550$, $4740$,
\\ && $4920$, $5622$, $6608$, $7372$, $7595$, $7783$, $7916$, $9625$, $10800$, $12125$, $12704$, $16275$, $19879$,
\\ && $20440$, $23625$, $26446$, $26684$, $29100$, $30472$, and 22 more.
\\ Block 2 & $2$ & $1953125$
\\ Block 3 & $1$ & $8$, $48$, $64$, $104$, $168_2$, $344$, $448_2$, $560$, $664$, $720$, $952$, $1008$, $1064$, $1624$, $2360$,
\\ &&  $2800$, $2912$, $2968$, $3752$,  $4200$, $4584$, $4600$, $4992$, $7112$, $8408$,
\\ && $8616$, $9416$, $14936$, $17080$, $28400$, and 20 more.
\\\hline\end{tabular}\end{center}

\subsection*{Highest weight labels}

These are only up to dimension 2000.

\begin{center}\begin{tabular}{cccccccccccccc}
\hline $\lambda$&000&100&010&200&002&300&110&020&102&400&004&210&012
\\\hline $L(\lambda)$ &$1$&$7$&$21$&$27$&$35$&$77$&$105$&$141$&$168_1$&$182$&$293$&$330$&$371_1$
\\ $\dim(W(\lambda))$&$1$&$7$&$21$&$27$&$35$&$77$&$105$&$168$&$189$&$182$&$294$&$330$&$378$
\\\hline $\lambda$&310&202&220&120&030&130&104&112&022&040&302&410&
\\\hline $L(\lambda)$&$371_2$&$448_1$& $483$&$616$&$825$&$988$&$1093$&$1246$&$1322$&$1513$&$1560$&$1750$&
\\ $\dim(W(\lambda))$&$819$&$616$&$1911$&$693$&$825$&$3003$&$1386$&$1617$&$2079$&$3003$&$1560$&$1750$&
\\\hline
\end{tabular}\end{center}

\begin{center}\begin{tabular}{cccccccc}
\hline $\lambda$&001&101&011&003&201&111&301
\\\hline $L(\lambda)$ &$8$&$48$&$64$&$104$&$168_2$&$344$&$448_2$
\\ $\dim(W(\lambda))$&$8$&$48$&$112$&$112$&$168$&$512$&$448$
\\\hline $\lambda$&103&013&021&311&401&211&203
\\\hline $L(\lambda)$&$560$&$664$&$720$&$952$& $1008$&$1064$&$1624$
\\ $\dim(W(\lambda))$&$560$&$1008$&$720$&$3584$&$1008$&$1512$&$1728$
\\\hline
\end{tabular}\end{center}


\subsection*{Extension information}


\begin{center}\begin{tabular}{ccc}\hline
Module & Tested to & $\Ext^1=k$
\\ \hline $1$ & $2380
 & $293$, $483$, $1322
\\ $7$ & $483$ & $371_1$, $371_2$
\\ $21$ & $182$ & $168_1$
\\ $27$ & $182$ & $141$
\\ $35$ & $182$ & None
\\ $77$ & $168$ & None
\\ $105$ & $141$ & None
\\ \hline\end{tabular}\end{center}

\begin{center}\begin{tabular}{ccc}\hline
Module & Tested to & $\Ext^1=k$
\\ \hline $8$ & $720$ & $104$, $664$
\\ $48$ & $168$ & $64$
\\ $64$ & $168$ & $48$
\\ $104$ & $104$ & $8$
\\ \hline\end{tabular}\end{center}

\newpage
\section{\texorpdfstring{$\Omega_7(7)$}{O7(7)}}

\subsection*{Block information}

These modules are only necessarily complete up to dimension 2000.

\begin{center}\begin{tabular}{ccc}
\hline Block & Kernel & Modules
\\\hline 1 & 2 & $1$, $7$, $21$, $26$, $35$, $77$, $105$, $168_1$, $182$, $189$, $294$, $304$, $378_1$, $378_2$, $521$, $616$, $693$, $714$,
\\ && $819$, $1008$, $1247$, $1256_1$, $1351$, $1463$, $1617$, $1750$, $1911$, $2030$, $2119$, $2121$, $2184$, \\ && $2821$, $3857$, $3927$, $4312$, $4654$, $5615$, $5698$, $5810$, $5985_1$, $5985_2$, $6538$, $6545$, $7931$,
\\ && $8904$, $10693$, $11704$, $14721$, $18627$, $20825$, and $145$ more
\\ 2 & 2 & $40353607$
\\ 3 & 1 & $8$, $40$, $112_1$, $112_2$, $168_2$, $248$, $448_1$, $448_2$, $472$, $560_1$, $560_2$, $1008$, $1256_2$, $1288$,
\\ && $1512$, $2080$, $2632$, $2912$, $3584$, $3808$, and $127$ more
\\\hline\end{tabular}\end{center}


\subsection*{Highest weight labels}

\begin{center}\begin{tabular}{cccccccccc}
\\\hline $\lambda$&000&100&010&200&002&300&110&020&400
\\ \hline $L(\lambda)$ & $1$&$7$&$21$&$26$&$35$&$77$&$105$&$168_1$&$182$
\\ $\dim(W(\lambda))$&$1$&$7$&$21$&$27$&$35$&$77$&$105$&$168$&$182$
\\\hline $\lambda$&102&004&210&500&012&030&202&120&600
\\\hline $L(\lambda)$&$189$&$294$&$304$&$378_1$&$378_2$&$521$&$616$&$693$&$714$
\\ $\dim(W(\lambda))$&$189$&$294$&$330$&$378$&$378$&$825$&$616$&$693$&$714$
\\\hline $\lambda$&310&104&510&302&006&022&112&410&220
\\\hline $L(\lambda)$&$819$&$1008$&$1247$&$1256_1$&$1351$&$1463$&$1617$&$1750$&$1911$
\\ $\dim(W(\lambda))$&$819$&$1386$&$3366$&$1560$&$1386$&$2079$&$1617$&$1750$&$1911$
\\ \hline
\end{tabular}\end{center}



\begin{center}\begin{tabular}{ccccccccc}
\hline $\lambda$&001&101&003&011&201&021&301&013
\\ \hline $L(\lambda)$ &$8$&$40$&$112_1$&$112_2$&$168_2$&$248$&$448_1$&$448_2$
\\ $\dim(W(\lambda))$&$8$&$48$&$112$&$112$&$168$&$720$&$448$&$1008$
\\ \hline $\lambda$&111&005&103&401&203&121&211&
\\ \hline $L(\lambda)$&$472$&$560_1$&$560_2$&$1008$&$1256_2$&$1288$&$1512$&
\\ $\dim(W(\lambda))$&$512$&$672$&$560$&$1008$&$1728$&$2800$&$1512$&
\\ \hline
\end{tabular}\end{center}

\subsection*{Extension information}


\begin{center}\begin{tabular}{ccc}\hline
Module & Tested to & $\Ext^1=k$
\\ \hline $1$ & $4312$ & $26$, $2030$
\\ $7$ & $521$ & None
\\ $21$ & $189$ & None
\\ $26$ & $189$ & $1$
\\ $35$ & $189$ & None
\\ $77$ & $168$ & None
\\ $105$ & $77$ & None
\\ \hline\end{tabular}
\end{center}

\newpage
\section{\texorpdfstring{$\Omega_9(3)$}{O9(3)}}

\subsection*{Block information}

These modules are only necessarily complete up to dimension 4000.

\begin{center}\begin{tabular}{ccc}
\hline Block & Kernel & Modules
\\\hline $1$ & $2$ & $1,9,36,43,84,126,147,369,579,594,840,867,1097,1422,2448,$
\\&&$2457,2736,3017,3060,7293,8748,11241,12393,14760,$...
\\ $2$ & $2$ & $43046721$
\\ $3$ & $1$ & $16,112,432,576,656,1776,2352,4464,6912,9504,17120,$...
\\\hline\end{tabular}\end{center}

\subsection*{Highest weight labels}

These are only up to dimension 4000.

\begin{center}\begin{tabular}{ccccccccccc}
\hline $\lambda$&0000&1000&0100&2000&0010&0002&1100&0200&0110&1010
\\\hline $L(\lambda)$ &$1$&$9$&$36$&$43$&$84$&$126$&$147$&$369$&$579$&$594$
\\ $\dim(W(\lambda))$&$1$&$9$&$36$&$44$&$84$&$126$&$231$&$495$&$1650$&$594$
\\\hline $\lambda$&1002&2100&0020&0012&1200&2010&0102&1110&2002&
\\\hline $L(\lambda)$&$840$&$867$&$1097$& $1422$&$2448$&$2457$&$2736$&$3017$&$3060$&
\\ $\dim(W(\lambda))$&$924$&$910$&$1980$&$4158$&$2574$&$2457$&$2772$&$9009$&$3900$&
\\\hline
\end{tabular}\end{center}

\begin{center}\begin{tabular}{cccccccc}
\hline $\lambda$&0001&1001&0101&2001&0011&1101&0201
\\\hline $L(\lambda)$&$16$&$112$&$432$&$576$&$656$&$1776$&$2352$
\\ $\dim(W(\lambda))$&$16$&$128$&$432$&$576$&$768$&$2560$&$4928$
\\\hline
\end{tabular}\end{center}

\subsection*{Extension information}

\begin{center}\begin{tabular}{ccc}\hline
Module & Tested to & $\Ext^1=k$
\\ \hline $1$ & $579$ & $43,369,579$
\\ $9$ & $84$ & None
\\ $36$ & $84$ & None
\\ $43$ & $84$ & $1$
\\ $84$ & $84$ & None
\\ \hline\end{tabular}\end{center}

\newpage
\section{\texorpdfstring{$\Omega_9(5)$}{O9(5)}}

\subsection*{Block information}

These modules are only necessarily complete up to dimension 4000.

\begin{center}\begin{tabular}{ccc}
\hline Block & Kernel & Modules
\\\hline $1$ & $2$ & $1,9,36,44,84,126,156,231,450,495,594,798,910,957,1485,1650,2094,2343,$
\\&&$2383,2719,3891,3968...$
\\ $2$ & $2$ & $152587890625$
\\ $3$ & $1$ & $16,128,336,432,544,576,1920,2480,2560,3360,$...
\\\hline\end{tabular}\end{center}


\subsection*{Highest weight labels}

These are only up to dimension 4000.

\begin{center}\begin{tabular}{c|cccccccccccc}
\hline $\lambda$&0000&1000&0100&2000&0010&0002&3000&1100&4000&0200&1010&1002
\\ \hline $L(\lambda)$ & $1$&$9$&$36$&$44$&$84$&$126$&$156$&$231$&$450$&$495$&$594$&$798$
\\ $\dim(W(\lambda))$&$1$&$9$&$36$&$44$&$84$&$126$&$156$&$231$&$450$&$495$&$594$&$924$
\\\hline $\lambda$&2100&3100&0020&0110&2010&0102&1200&2200&0004&0012&0300&
\\\hline $L(\lambda)$ & $910$&$957$&$1485$&$1650$&$1659$&$2094$&$2343$&$2383$&$2719$&$3891$&$3968$&
\\ $\dim(W(\lambda))$&$910$&$2772$&$1980$&$1650$&$2457$&$2772$&$2574$&$8748$&$2772$&$4158$&$4004$&
\\ \hline
\end{tabular}\end{center}

\begin{center}\begin{tabular}{c|cccccccccc}
\hline $\Lambda$&0001&1001&0011&0101&0003&2001&3001&1011&1101&4001
\\ \hline $L(\lambda)$&$16$&$128$&$336$&$432$&$544$&$576$&$1920$&$2480$&$2560$&$3360$
\\ $\dim(W(\lambda))$&$16$&$128$&$768$&$432$&$672$&$576$&$1920$&$5040$&$2560$&$5280$
\\ \hline
\end{tabular}\end{center}


\subsection*{Extension information}


\begin{center}\begin{tabular}{ccc}\hline
Module & Tested to & $\Ext^1=k$
\\ \hline $1$ & $1659$ & None
\\ $9$ & $231$ & None
\\ $36$ & $84$ & None
\\ $44$ & $84$ & None
\\ \hline\end{tabular}\end{center}

\newpage
\section{\texorpdfstring{$\Omega_8^+(2)$}{O8+(2)}}

\subsection*{Block information}

\begin{center}\begin{tabular}{ccc}
\hline Block & Kernel & Modules
\\\hline $1$ & $1$ & $1,8_1,8_2,8_3,26,48_1,48_2,48_3,160_1,160_2,160_3,246,784_1,784_2,784_3$
\\ $2$ & $1$ & $4096$
\\\hline\end{tabular}\end{center}

\subsection*{Identification of modules}

Since there is an $S_3$ of automorphisms permuting the $8$-dimensional module, we can make any choice for $8_1$, $8_2$ and $8_3$. We then define $8_i\otimes 26=48_i\oplus 160_i$, and $748_i$ as the socle of $8_i\otimes 246$.


\subsection*{Highest weight labels}

\begin{center}\begin{tabular}{ccccccccc}
\hline $\lambda$&0000&1000&0010&0001&0100&0011&1001&1010
\\\hline $L(\lambda)$ &$1$&$8_1$&$8_2$&$8_3$&$26$&$48_1$&$48_2$&$48_3$
\\ $\dim(W(\lambda))$&$1$&$8$&$8$&$8$&$28$&$56$&$56$&$56$
\\\hline $\lambda$&1100&0110&0101&1011&0111&1101&1110&1111
\\\hline $L(\lambda)$&$160_1$&$160_2$&$160_3$&$246$&$784_1$&$784_2$&$784_3$&$4096$
\\ $\dim(W(\lambda))$&$160$&$160$&$160$&$350$&$840$&$840$&$840$&$4096$
\\\hline
\end{tabular}\end{center}


\subsection*{Extension information}

\begin{center}\begin{tabular}{c|ccccccccccccccc}
Block 1 &$1$&$8_1$&$8_2$&$8_3$&$26$&$48_1$&$48_2$&$48_3$&$160_1$&$160_2$&$160_3$&$246$&$784_1$&$784_2$&$784_3$
\\ \hline $1$&$0$&$0$&$0$&$0$&$2$&$1$&$1$&$1$&$0$&$0$&$0$&$1$&$0$&$0$&$0$
\\$8_1$&$0$&$0$&$0$&$0$&$1$&$1$&$0$&$0$&$0$&$1$&$1$&$0$&$1$&$0$&$0$
\\$8_2$&$0$&$0$&$0$&$0$&$1$&$0$&$1$&$0$&$1$&$0$&$1$&$0$&$0$&$1$&$0$
\\$8_3$&$0$&$0$&$0$&$0$&$1$&$0$&$0$&$1$&$1$&$1$&$0$&$0$&$0$&$0$&$1$
\\$26$&$2$&$1$&$1$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$2$&$0$&$0$&$0$
\\$48_1$&$1$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\$48_2$&$1$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\$48_3$&$1$&$0$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\$160_1$&$0$&$0$&$1$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\$160_2$&$0$&$1$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\$160_3$&$0$&$1$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\$246$&$1$&$0$&$0$&$0$&$2$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\$784_1$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\$784_2$&$0$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\$784_3$&$0$&$0$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\end{tabular}\end{center}


\subsection*{Symmetric and exterior squares of modules}

\begin{center}\begin{tabular}{c|cc}
Block 1 &$8_i$&$26$
\\\hline $S^2(M)$&$1/26/1,8_i$&$1\oplus (26/1,1,8_1,8_2,8_3,246/8,8)$
\\ $\Lambda^2(M)$&$1/26/1$&$1\oplus (26/1,1,8_1,8_2,8_3,246/8)$
\end{tabular}\end{center}

\subsection*{Actions of unipotent elements on modules}

\begin{center}\begin{tabular}{c|cccccccc}
Block 1 &$1$&$8_1$&$8_2$&$8_3$&$26$&$48_1$&$48_2$&$48_3$
\\\hline 2A&$1$&$2^2,1^4$&$2^2,1^4$&$2^2,1^4$&$2^{10},1^6$&$2^{20},1^8$&$2^{20},1^8$&$2^{20},1^8$
\\2B&$1$&$2^2,1^4$&$2^4$&$2^4$&$2^{10},1^6$&$2^{20},1^8$&$2^{24}$&$2^{24}$
\\ 2C&$1$&$2^4$&$2^2,1^4$&$2^4$&$2^{10},1^6$&$2^{24}$&$2^{20},1^8$&$2^{24}$
\\2D&$1$&$2^4$&$2^4$&$2^2,1^4$&$2^{10},1^6$&$2^{24}$&$2^{24}$&$2^{20},1^8$
\\2E&$1$&$2^4$&$2^4$&$2^4$&$2^{12},1^2$&$2^{24}$&$2^{24}$&$2^{24}$
\\4A[A],4B[A]&$1$&$3^2,1^2$&$3^2,1^2$&$3^2,1^2$&$4^2,3^6$&$4^8,3^4,1^4$&$4^8,3^4,1^4$&$4^8,3^4,1^4$
\\4C[B]&$1$&$4,2,1^2$&$4^2$&$4^2$&$4^5,2^3$&$4^{10},2^4$&$4^{12}$&$4^{12}$
\\4D[C]&$1$&$4^2$&$4,2,1^2$&$4^2$&$4^5,2^3$&$4^{12}$&$4^{10},2^4$&$4^{12}$
\\4E[D]&$1$&$4^2$&$4^2$&$4,2,1^2$&$4^5,2^3$&$4^{12}$&$4^{12}$&$4^{10},2^4$
\\4F[E]&$1$&$4^2$&$4^2$&$4^2$&$4^6,1^2$&$4^{12}$&$4^{12}$&$4^{12}$
\\8A[A],8B[B]& $1$&$6,2$&$6,2$&$6,2$&$8,6^3$&$8^4,6^2,2^2$&$8^4,6^2,2^2$&$8^4,6^2,2^2$
\end{tabular}\end{center}


\begin{center}\begin{tabular}{c|ccccccc}
Block 1&$160_1$&$160_2$&$160_3$&$246$&$784_1$&$784_2$&$784_3$
\\ \hline 2A&$2^{72},1^{16}$&$2^{72},1^{16}$&$2^{72},1^{16}$&$2^{118},1^{10}$&$2^{388},1^8$&$2^{388},1^8$&$2^{388},1^8$
\\2B&$2^{72},1^{16}$&$2^{80}$&$2^{80}$&$2^{118},1^{10}$&$2^{388},1^8$&$2^{392}$&$2^{392}$
\\2C&$2^{80}$&$2^{72},1^{16}$&$2^{80}$&$2^{118},1^{10}$&$2^{392}$&$2^{388},1^8$&$2^{392}$
\\2D&$2^{80}$&$2^{80}$&$2^{72},1^{16}$&$2^{118},1^{10}$&$2^{392}$&$2^{392}$&$2^{388},1^8$
\\2E&$2^{80}$&$2^{80}$&$2^{80}$&$2^{120},1^6$&$2^{392}$&$2^{392}$&$2^{392}$
\\4A[A],4B[A]&$4^{32},3^8,1^8$&$4^{32},3^8,1^8$&$4^{32},3^8,1^8$&$4^{56},3^6,2^2$&$4^{192},3^4,1^4$&$4^{192},3^4,1^4$&$4^{192},3^4,1^4$
\\4C[B]&$4^{36},2^8$&$4^{40}$&$4^{40}$&$4^{58},3^2,2^4$&$4^{194},2^4$&$4^{196}$&$4^{196}$
\\4D[C]&$4^{40}$&$4^{36},2^8$&$4^{40}$&$4^{58},3^2,2^4$&$4^{196}$&$4^{194},2^4$&$4^{196}$
\\4E[D]&$4^{40}$&$4^{40}$&$4^{36},2^8$&$4^{58},3^2,2^4$&$4^{196}$&$4^{196}$&$4^{194},2^4$
\\4F[E]&$4^{40}$&$4^{40}$&$4^{40}$&$4^{60},2^2,1^2$&$4^{196}$&$4^{196}$&$4^{196}$
\\ 8A[A],8B[B]&$8^{16},6^4,2^4$&$8^{16},6^4,2^4$&$8^{16},6^4,2^4$&$8^{28},6^3,4$&$8^{96},6^2,2^2$&$8^{96},6^2,2^2$&$8^{96},6^2,2^2$
\end{tabular}\end{center}


\subsection*{Traces of semisimple elements on modules}

\begin{small}\begin{center}\begin{tabular}{c|cccccccccccccccc}
Block 1,2 &$1$&$8_1$&$8_2$&$8_3$&$26$&$48_1$&$48_2$&$48_3$&$160_1$&$160_2$&$160_3$&$246$&$784_1$&$784_2$&$784_3$&$4096$
\\\hline 3A&$1$&$5$&$-4$&$-4$&$8$&$6$&$-12$&$-12$&$34$&$-20$&$-20$&$12$&$10$&$-44$&$-44$&$64$
\\ 3B&$1$&$-4$&$5$&$-4$&$8$&$-12$&$6$&$-12$&$-20$&$34$&$-20$&$12$&$-44$&$10$&$-44$&$64$
\\ 3C&$1$&$-4$&$-4$&$5$&$8$&$-12$&$-12$&$6$&$-20$&$-20$&$34$&$12$&$-44$&$-44$&$10$&$64$
\\ 3D&$1$&$-1$&$-1$&$-1$&$-1$&$3$&$3$&$3$&$-2$&$-2$&$-2$&$3$&$1$&$1$&$1$&$-8$
\\3E&$1$&$2$&$2$&$2$&$-1$&$0$&$0$&$0$&$-2$&$-2$&$-2$&$-6$&$-8$&$-8$&$-8$&$-8$
\\5A&$1$&$3$&$-2$&$-2$&$1$&$-2$&$-2$&$-2$&$5$&$0$&$0$&$-4$&$-6$&$4$&$4$&$-4$
\\5B&$1$&$-2$&$3$&$-2$&$1$&$-2$&$-2$&$-2$&$0$&$5$&$0$&$-4$&$4$&$-6$&$4$&$-4$
\\5C&$1$&$-2$&$-2$&$3$&$1$&$-2$&$-2$&$-2$&$0$&$0$&$5$&$-4$&$4$&$4$&$-6$&$-4$
\\7A&$1$&$1$&$1$&$1$&$-2$&$-1$&$-1$&$-1$&$-1$&$-1$&$-1$&$1$&$0$&$0$&$0$&$1$
\\9A[D]&$1$&$2$&$-1$&$-1$&$-1$&$-3$&$0$&$0$&$1$&$1$&$1$&$0$&$1$&$1$&$1$&$1$
\\9B[D]&$1$&$-1$&$2$&$-1$&$-1$&$0$&$-3$&$0$&$1$&$1$&$1$&$0$&$1$&$1$&$1$&$1$
\\9C[D]&$1$&$-1$&$-1$&$2$&$-1$&$0$&$0$&$-3$&$1$&$1$&$1$&$0$&$1$&$1$&$1$&$1$
\\15A[A,A]&$1$&$0$&$1$&$1$&$-2$&$1$&$-2$&$-2$&$-1$&$0$&$0$&$2$&$0$&$1$&$1$&$-1$
\\15B[B,B]&$1$&$1$&$0$&$1$&$-2$&$-2$&$1$&$-2$&$0$&$-1$&$0$&$2$&$1$&$0$&$1$&$-1$
\\15C[C,C]&$1$&$1$&$1$&$0$&$-2$&$-2$&$-2$&$1$&$0$&$0$&$-1$&$2$&$1$&$1$&$0$&$-1$
\end{tabular}\end{center}\end{small}

\newpage
\section{\texorpdfstring{$\Omega_8^+(3)$}{O8+(3)}}

\subsection*{Block information}

\begin{center}\begin{tabular}{ccc}
\hline Block & Kernel & Modules
\\\hline $1$ & $2\times 2$ & $1$, $28$, $35_1$, $35_2$, $35_3$, $195$, $322$, $518_1$, $518_2$, $518_3$, $567_1$, $567_2$, $567_3$, $1841$,
\\&& $3948_1$, $3948_2$, $3948_3$, $8132$, $8343_1$, $8343_2$, $8343_3$, $13776$, $43254_1$, $43254_2$, $43254_3$, $96753$
\\ $2$ & $2\times 2$ & $531441$
\\ $3$ & $2_1$ & $8_1$, $56_1$, $104_1$, $224_1$, $224_2$, $384_1$, $1176_1$, $1296_1$, $1896_1$, $1896_2$,
\\ && $3240_1$, $4768_1$, $7600_1$, $18144_1$, $18144_2$, $24240_1$, $58920_1$, $196344_1$
\\ $4$ & $2_2$ & $8_2$, $56_2$, $104_2$, $224_3$, $224_4$, $384_2$, $1176_2$, $1296_2$, $1896_3$, $1896_4$
\\ && $3240_2$, $4768_2$, $7600_2$, $18144_3$, $18144_4$, $24240_2$, $58920_2$, $196344_2$
\\ $5$ & $2_3$ & $8_3$, $56_3$, $104_3$, $224_5$, $224_6$, $384_3$, $1176_3$, $1296_3$, $1896_5$, $1896_6$,
\\ && $3240_3$, $4768_3$, $7600_3$, $18144_5$, $18144_6$, $24240_3$, $58920_3$, $196344_3$
\\\hline\end{tabular}\end{center}

\subsection*{Identification of modules}

Since there is an $S_3$ of automorphisms permuting the $8$-dimensional module, we can make any choice for $8_1$, $8_2$ and $8_3$. Then $35_i$ lies in $S^2(8_i)$, $518_{i+2}$ lies in $35_i\otimes 35_{i+1}$ (with the indices taken modulo $3$), and $567_i$ lies in $28\otimes 35_i$. Finally, $3948_i$ is a composition factor of $35_i\otimes 195$.

\subsection*{Highest weight labels}

\noindent Blocks 1 and 2:
\begin{center}\begin{tabular}{cccccccccc}
\hline $\lambda$&0000&0100&2000&0020&0002&0200&1011&0022&2002
\\\hline $L(\lambda)$ &$1$&$28$&$35_1$&$35_2$&$35_3$&$195$&$322$&$518_1$&$518_2$
\\ $\dim(W(\lambda))$&$1$&$28$&$35$&$35$&$35$&$300$&$350$&$840$&$840$
\\\hline $\lambda$&2020&2100&0120&0102&1111&2200&0220&0202&2022
\\\hline $L(\lambda)$&$518_3$&$567_1$&$567_2$&$567_3$&$1841$&$3948_1$&$3948_2$&$3948_3$&$8132$
\\ $\dim(W(\lambda))$&$840$&$567$&$567$&$567$&$4096$&$4312$&$4312$&$4312$&$15092$
\\\hline $\lambda$&2120&2102&0122&1211&2220&2202&0222&2122&2222
\\\hline $L(\lambda)$&$8343_1$&$8343_2$&$8343_3$&$13776$&$43254_1$&$43254_2$&$43254_3$&$96753$&$531441$
\\ $\dim(W(\lambda))$&$8910$&$8910$&$8910$&$25725$&$51597$&$51597$&$51597$&$114400$&$531441$
\\\hline
\end{tabular}\end{center}

\noindent Block 3:

\begin{center}\begin{tabular}{cccccccccc}
\hline $\lambda$&1000&0011&1100&1020&1002&0111&1200&2011&1102
\\\hline $L(\lambda)$ &$8_1$&$56_1$&$104_1$&$224_1$&$224_2$&$384_1$&$1176_1$&$1296_1$&$1896_1$
\\ $\dim(W(\lambda))$&$8$&$56$&$160$&$224$&$224$&$840$&$1400$&$1296$&$2800$
\\\hline $\lambda$&1120&1022&0211&2111&1220&1202&1122&2211&1222
\\\hline $L(\lambda)$&$1896_2$&$3240_1$&$4768_1$&$7600_1$&$18144_1$&$18144_2$&$24240_1$&$58920_1$&$196344_1$
\\ $\dim(W(\lambda))$&$2800$&$4536$&$6160$&$12936$&$18144$&$18144$&$39200$&$72800$&$199564$
\\\hline
\end{tabular}\end{center}

\noindent Block 4:

\begin{center}\begin{tabular}{cccccccccc}
\hline $\lambda$&0010&1001&0110&0012&2010&1101&0210&1021&2110
\\\hline $L(\lambda)$ &$8_2$&$56_2$&$104_2$&$224_2$&$224_2$&$384_2$&$1176_2$&$1296_2$&$1896_2$
\\ $\dim(W(\lambda))$&$8$&$56$&$160$&$224$&$224$&$840$&$1400$&$1296$&$2800$
\\\hline $\lambda$&0112&2012&1201&1121&2210&0212&2112&1221&2212
\\\hline $L(\lambda)$&$1896_2$&$3240_2$&$4768_2$&$7600_2$&$18144_2$&$18144_2$&$24240_2$&$58920_2$&$196344_2$
\\ $\dim(W(\lambda))$&$2800$&$4536$&$6160$&$12936$&$18144$&$18144$&$39200$&$72800$&$199564$
\\\hline
\end{tabular}\end{center}

\noindent Block 5:

\begin{center}\begin{tabular}{cccccccccc}
\hline $\lambda$&0001&1010&0101&2001&0021&1110&0201&1012&2101
\\\hline $L(\lambda)$ &$8_3$&$56_3$&$104_3$&$224_3$&$224_3$&$384_3$&$1176_3$&$1296_3$&$1896_3$
\\ $\dim(W(\lambda))$&$8$&$56$&$160$&$224$&$224$&$840$&$1400$&$1296$&$2800$
\\\hline $\lambda$&0121&2021&1210&1112&0221&2201&2121&1212&2221
\\\hline $L(\lambda)$&$1896_3$&$3240_3$&$4768_3$&$7600_3$&$18144_3$&$18144_3$&$24240_3$&$58920_3$&$196344_3$
\\ $\dim(W(\lambda))$&$2800$&$4536$&$6160$&$12936$&$18144$&$18144$&$39200$&$72800$&$199564$
\\\hline
\end{tabular}\end{center}

\subsection*{Extension information}


\begin{center}\begin{tabular}{c|ccccccccccccccc}
Block 1 & $1$&$28$&$35_1$&$35_2$&$35_3$&$195$&$322$&$518_1$&$518_2$&$518_3$&$567_1$&$567_2$&$567_3$&$1841$&$3948_i$
\\\hline $1$ &$0$&$0$&$0$&$0$&$0$&$3$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$1$&$1$
\\ $28$&$0$&$0$&$0$&$0$&$0$&$3$&&&&&&&&&
\\ $35_1$&$0$&$0$&$0$&$0$&$0$&$2$&&&&&&&&&
\end{tabular}\end{center}

\newpage
\section{\texorpdfstring{$\Omega_8^+(5)$}{O8+(5)}}

\subsection*{Block information}
These modules are only necessarily complete up to dimension 10000.
\begin{center}\begin{tabular}{ccc}
\hline Block & Kernel & Modules
\\\hline $1$ & $2\times 2$ & $1,28,35_1,35_2,35_3,293_1,293_2,293_3,300,350,539_1,539_2,539_3,805_1,805_2,805_3,$
\\ &&$1322_1,1322_2,1322_3,1925,2451,2501,3403_1,3403_2,3403_3,3675_1,3675_2,3675_3,$
\\ &&$5775_1,5775_2,5775_3,5775_4,5775_5,5775_6,7372_1,7372_2,7372_3,8105_1,8105_2,8105_3,$... 
\\ $2$ & $2\times 2$ & $244140625$
\\ $3$ & $2$ & $8_1,56_1,104_1,160_1,168_1,168_2,664_1,664_2,664_3,680_1,904_1,1400_1,$
\\ &&$1680_1,1680_2,2360_1,2400_1,2400_2,2800_1,2800_2,3096_1,3632_1,$
\\ && $3632_2,4992_1,5480_1,6600_1,6944_1,6944_2,7552_1,8800_1,9416_1,$...
\\ $4$ & $2$ & Image of block 3 under outer automorphism
\\ $5$ & $2$ & Image of block 3 under outer automorphism
\\\hline\end{tabular}\end{center}


\subsection*{Highest weight labels}

\begin{center}\begin{tabular}{cccccccccc}
\hline $\lambda$&0000&0100&2000&0020&0002&4000&0040&0004&0200
\\\hline$L(\lambda)$ &$1$&$28$&$35_1$&$35_2$&$35_3$&$293_1$&$293_2$&$293_3$&$300$
\\ $\dim(W(\lambda))$&$1$&$28$&$35$&$35$&$35$&$294$&$294$&$294$&$300$
\\\hline $\lambda$&1011&2100&1200&0102&2020&2002&0022&2200&0220
\\\hline$L(\lambda)$ &$350$&$539_1$&$539_2$&$539_3$&$805_1$&$805_2$&$805_3$&$1322_1$&$1322_2$
\\ $\dim(W(\lambda))$&$350$&$567$&$567$&$567$&$840$&$840$&$840$&$4312$&$4312$
\\\hline $\lambda$&0202&0300&1111&0400&4100&0140&0104&3011&1031
\\\hline$L(\lambda)$ &$1322_3$&$1925$&$2451$&$2501$&$3403_1$&$3403_2$&$3403_3$&$3675_1$&$3675_2$
\\ $\dim(W(\lambda))$&$4312$&$1925$&$4096$&$8918$&$3696$&$3696$&$3696$&$3675$&$3675$
\\\hline $\lambda$&1013&4020&4002&2040&2004&0042&0024&2220&2202
\\\hline$L(\lambda)$&$3675_3$&$5775_1$&$5775_2$&$5775_3$&$5775_4$&$5775_5$&$5775_6$&$7372_1$&$7373_2$
\\ $\dim(W(\lambda))$&$3675$&$5775$&$5775$&$5775$&$5775$&$5775$&$5775$&$51597$&$51597$
\\\hline $\lambda$&0222&2120&2102&0122&&&&&
\\\hline$L(\lambda)$&$7373_3$&$8105_1$&$8105_2$&$8105_3$&&&&&
\\ $\dim(W(\lambda))$&$51597$&$8910$&$8910$&$8910$&&&&&
\\ \hline
\end{tabular}\end{center}

\begin{center}\begin{tabular}{ccccccccc}
\hline $\lambda$&1000&0011&3000&1100&1020&1002&3100&0031
\\\hline $L(\lambda)$&$8_1$&$56_1$&$104_1$&$160_1$&$168_1$&$168_2$&$664_1$&$664_2$
\\ $\dim(W(\lambda))$&$8$&$56$&$112$&$160$&$224$&$224$&$1568$&$672$
\\\hline $\lambda$&0013&0111&2011&1200&1040&1004&1300&3020
\\\hline $L(\lambda)$&$664_3$&$680_1$&$904_1$&$1400_1$&$1680_1$&$1680_2$&$2360_1$&$2400_1$
\\ $\dim(W(\lambda))$&$672$&$840$&$1296$&$1400$&$1680$&$1680$&$7480$&$2400$
\\\hline $\lambda$&3002&1120&1102&1022&0131&0113&0311&0211
\\\hline $L(\lambda)$&$2400_2$&$2800_1$&$2800_2$&$3096_1$&$3632_1$&$3632_2$&$4992_1$&$5480_1$
\\ $\dim(W(\lambda))$&$2400$&$2800$&$2800$&$4536$&$7392$&$7392$&$30576$&$6160$
\\\hline $\lambda$&0033&1220&1202&3200&4011&0133&&
\\\hline $L(\lambda)$&$6600_1$&$6944_1$&$6944_2$&$7552_1$&$8800_1$&$9416_1$&&
\\ $\dim(W(\lambda))$&$6600$&$18144$&$18144$&$10752$&$8800$&$56056$&&
\\\hline
\end{tabular}\end{center}

\subsection*{Extension information}

\begin{center}\begin{tabular}{ccc}\hline
Module & Tested to & $\Ext^1=k$
\\\hline $1$ & $539$ & $293$
\\ $28$ & $35$ & None
\\ $35_i$ & $35$ & None
\\ \hline\end{tabular}\end{center}

\subsection*{Actions of unipotent elements on modules}

\begin{center}\begin{tabular}{c|ccccc}
Block 1&$1$&$28$&$35_1$&$35_2$&$35_3$
\\\hline 5A&$1$&$3,2^8,1^9$&$3^3,2^8,1^{10}$&$3^3,2^8,1^{10}$&$3^3,2^8,1^{10}$
\\5B,5C&$1$&$3^6,1^{10}$&$5,3^5,1^{15}$&$3^{10},1^5$&$3^{10},1^5$
\\5D,5E&$1$&$3^6,1^{10}$&$3^{10},1^5$&$5,3^5,1^{15}$&$3^{10},1^5$
\\5F,5G&$1$&$3^6,1^{10}$&$3^{10},1^5$&$3^{10},1^5$&$5,3^5,1^{15}$
\\5H,5I,5J,5K&$1$&$4^2,3^3,2^4,1^3$&$5,4^2,3^4,2^4,1^2$&$5,4^2,3^4,2^4,1^2$&$5,4^2,3^4,2^4,1^2$
\\5L,5M&$1$&$5,3^7,1^2$&$5^3,3^5,1^5$&$5^3,3^5,1^5$&$5^3,3^5,1^5$
\\5N,5O&$1$&$5^5,1^3$&$5^6,1^5$&$5^7$&$5^7$
\\5P,5Q&$1$&$5^5,1^3$&$5^7$&$5^6,1^5$&$5^7$
\\5R,5S&$1$&$5^5,1^3$&$5^7$&$5^7$&$5^6,1^5$
\\5T,5U,5V,5W&$1$&$5^5,3$&$5^7$&$5^7$&$5^7$
\\25A,25B,25C,25D[A]&$1$&$11,7^2,3$&$13,9,7,5,1$&$13,9,7,5,1$&$13,9,7,5,1$
\end{tabular}\end{center}


\newpage
\section{\texorpdfstring{$\Omega_8^-(2)$}{O8-(2)}}

\subsection*{Block information}

\begin{center}\begin{tabular}{ccc}
\hline Block & Kernel & Modules
\\\hline $1$ & $1$ & $1,8_1,8_2,8_3,26,48_1,48_2,48_3,160_1,160_2,160_3,246,784_1,784_2,784_3$
\\ $2$ & $1$ & $4096$
\\\hline\end{tabular}\end{center}

\subsection*{Identification of modules}

Define $8_1$, $48_1$, $160_1$, $784_1$ as the modules defined over $\F_2$. For the modules $n_2$ and $n_3$ for the various $n$, we chose $8_2$ and $8_3$ arbitrarily, since they are permuted by the outer automorphism, and define the rest by $8_i\otimes 26=48_i\oplus 160_i$ and $784_i=\soc(8_i\otimes 246)$, as with $\Omega_8^+(2)$.

\subsection*{Extension information}


\begin{center}\begin{tabular}{c|ccccccccccccccc}
Block 1 &$1$&$8_1$&$8_2$&$8_3$&$26$&$48_1$&$48_2$&$48_3$&$160_1$&$160_2$&$160_3$&$246$&$784_1$&$784_2$&$784_3$
\\ \hline $1$&$0$&$0$&$0$&$0$&$2$&$1$&$0$&$0$&$0$&$1$&$1$&$1$&$0$&$0$&$0$
\\$8_1$&$0$&$0$&$0$&$0$&$1$&$1$&$1$&$1$&$0$&$0$&$0$&$0$&$1$&$0$&$0$
\\$8_2$&$0$&$0$&$0$&$0$&$1$&$0$&$1$&$0$&$1$&$0$&$1$&$0$&$0$&$1$&$0$
\\$8_3$&$0$&$0$&$0$&$0$&$1$&$0$&$0$&$1$&$1$&$1$&$0$&$0$&$0$&$0$&$1$
\\$26$&$2$&$1$&$1$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$2$&$0$&$0$&$0$
\\$48_1$&$1$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\$48_2$&$0$&$1$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\$48_3$&$0$&$1$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\$160_1$&$0$&$0$&$1$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\$160_2$&$1$&$0$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&&&$0$&$0$&$0$&$0$
\\$160_3$&$1$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&&&$0$&$0$&$0$&$0$
\\$246$&$1$&$0$&$0$&$0$&$2$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\$784_1$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\$784_2$&$0$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\$784_3$&$0$&$0$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\end{tabular}\end{center}


\subsection*{Actions of unipotent elements on modules}

\begin{center}\begin{tabular}{c|cccccccc}
Block 1 &$1$&$8_1$&$8_2$&$8_3$&$26$&$48_1$&$48_2$&$48_3$
\\\hline 2A&$1$&$2^2,1^4$&$2^2,1^4$&$2^2,1^4$&$2^{10},1^6$&$2^{20},1^8$&$2^{20},1^8$&$2^{20},1^8$
\\2B&$1$&$2^2,1^4$&$2^4$&$2^4$&$2^{10},1^6$&$2^{20},1^8$&$2^{24}$&$2^{24}$
\\2C&$1$&$2^4$&$2^4$&$2^4$&$2^{12},1^2$&$2^{24}$&$2^{24}$&$2^{24}$
\\4A[A],4B[A]&$1$&$3^2,1^2$&$3^2,1^2$&$3^2,1^2$&$4^2,3^6$&$4^8,3^4,1^4$&$4^8,3^4,1^4$&$4^8,3^4,1^4$
\\4C[B]&$1$&$4,2,1^2$&$4^2$&$4^2$&$4^5,2^3$&$4^{10},2^4$&$4^{12}$&$4^{12}$
\\4D[C]&$1$&$4^2$&$4^2$&$4^2$&$4^6,1^2$&$4^{12}$&$4^{12}$&$4^{12}$
\\ 8A[A],8B[B]&$1$&$6,2$&$6,2$&$6,2$&$8,6^3$&$8^4,6^2,2^2$&$8^4,6^2,2^2$&$8^4,6^2,2^2$
\end{tabular}\end{center}

\begin{center}\begin{tabular}{c|ccccccc}
Block 1&$160_1$&$160_2$&$160_3$&$246$&$784_1$&$784_2$&$784_3$
\\ \hline 2A&$2^{72},1^{16}$&$2^{72},1^{16}$&$2^{72},1^{16}$&$2^{118},1^{10}$&$2^{388},1^8$&$2^{388},1^8$&$2^{388},1^8$
\\2B&$2^{72},1^{16}$&$2^{80}$&$2^{80}$&$2^{118},1^{10}$&$2^{388},1^8$&$2^{392}$&$2^{392}$
\\2C&$2^{80}$&$2^{80}$&$2^{80}$&$2^{120},1^6$&$2^{392}$&$2^{392}$&$2^{392}$
\\4A[A],4B[A]&$4^{32},3^8,1^8$&$4^{32},3^8,1^8$&$4^{32},3^8,1^8$&$4^{56},3^6,2^2$&$4^{192},3^4,1^4$&$4^{192},3^4,1^4$&$4^{192},3^4,1^4$
\\4C[B]&$4^{36},2^8$&$4^{40}$&$4^{40}$&$4^{58},3^2,2^4$&$4^{194},2^4$&$4^{196}$&$4^{196}$
\\4D[C]&$4^{40}$&$4^{40}$&$4^{40}$&$4^{60},2^2,1^2$&$4^{196}$&$4^{196}$&$4^{196}$
\\ 8A[A],8B[B]&$8^{16},6^4,2^4$&$8^{16},6^4,2^4$&$8^{16},6^4,2^4$&$8^{28},6^3,4$&$8^{96},6^2,2^2$&$8^{96},6^2,2^2$&$8^{96},6^2,2^2$
\end{tabular}\end{center}

\subsection*{Traces of semisimple elements on modules}



\begin{small}\begin{center}\begin{tabular}{c|cccccccccccccccc}
Block 1,2 &$1$&$8_1$&$8_2$&$8_3$&$26$&$48_1$&$48_2$&$48_3$&$160_1$&$160_2$&$160_3$&$246$&$784_1$&$784_2$&$784_3$&$4096$
\\\hline 3A&$1$&$5$&$-4$&$-4$&$8$&$6$&$-12$&$-12$&$34$&$-20$&$-20$&$12$&$10$&$-44$&$-44$&$64$
\\3B&$1$&$2$&$2$&$2$&$-1$&$0$&$0$&$0$&$-2$&$-2$&$-2$&$-6$&$-8$&$-8$&$-8$&$-8$
\\ 3C&$1$&$-1$&$-1$&$-1$&$-1$&$3$&$3$&$3$&$-2$&$-2$&$-2$&$3$&$1$&$1$&$1$&$-8$
\\5A&$1$&$3$&$-2$&$-2$&$1$&$-2$&$-2$&$-2$&$5$&$0$&$0$&$-4$&$-6$&$4$&$4$&$-4$
\\7A&$1$&$1$&$1$&$1$&$-2$&$-1$&$-1$&$-1$&$-1$&$-1$&$-1$&$1$&$0$&$0$&$0$&$1$
\\9A[C]&$1$&$2$&$-1$&$-1$&$-1$&$-3$&$0$&$0$&$1$&$1$&$1$&$0$&$1$&$1$&$1$&$1$
\end{tabular}\end{center}\end{small}

\newpage
\section{\texorpdfstring{$\Omega_8^-(3)$}{O8-(3)}}

\subsection*{Block information}

\begin{center}\begin{tabular}{ccc}
\hline Block & Kernel & Modules
\\\hline $1$ & $2$ & $1,8_1,28,35_1,35_2,35_3,56_1,104_1,195,224_1,224_2,322,384_1,518_1,518_2,518_3$,
\\ && $567_1,567_2,567_3,1176_1,1296_1,1841,1896_1,1896_2,3240_1$,
\\&& $3948_1,3948_2,3948_3,4768_1,7600_1,8132,8343_1,8343_2,8343_3,13776$,
\\ && $18144_1,18144_2,24240_1,43254_1,43254_2,43254_3,58920_1,96753,196344_1$
\\ $2$ & $2$ &$ 531441$
\\ $3$ & $1$ & $8_2,8_3,56_2,56_3,104_2,104_3,224_3,224_4,224_5,224_6,384_2,384_3,1176_2,1176_3,1296_2,1296_3,$
\\&&$1896_3,1896_4,1896_5,1896_6,3240_2,3240_3,4768_,4768_3,7600_2,7600_3,18144_3,18144_4$,
\\ &&$18144_5,18144_6,24240_2,24240_3,58920_2,58920_3,196344_2,196344_3$
\\\hline\end{tabular}\end{center}


\subsection*{Extension information}
\begin{center}\begin{tabular}{cccc}
\hline Module & Tested to & $\Ext^1=k$ & $\Ext^1=k^2$
\\\hline 1 & $567$ & $195$ & $384$
\\$8$ & $195$ & $104$ & $195$
\\$28$ & $56$ & None & None
\\$35_1$ & $56$ & None & None
\\$35_2$ & $56$ & None & None
\\$56$ & $56$ & None & None
\\\hline\end{tabular}\end{center}


\newpage
\section{\texorpdfstring{$\Omega_8^-(5)$}{O8-(5)}}

\subsection*{Block information}
\begin{center}\begin{tabular}{ccc}
\hline Block & Kernel & Modules
\\\hline $1$ & $2$ & $1,8_1,28,35_1,35_2,35_3,56_1,104_1,160_1,168_1,168_2,293_1,293_2,293_3,300,350$,
\\&&$539_1,539_2,539_3,664_1,664_2,664_3,680_1,805_1,805_2,805_3,904_1$...
\\ $2$ & $2$ & $244140625$
\\ $3$ & $1$ & $8_2,8_3,56_2,56_3,104_2,104_3,160_2,160_3,168_3,168_4,168_5,168_6,$
\\ &&$664_4,664_5,664_6,664_7,664_8,664_9,680_2,680_3,904_2,904_3$...
\\\hline\end{tabular}\end{center}

\subsection*{Extension information}
\begin{center}\begin{tabular}{ccc}\hline
Module & Tested to & $\Ext^1=k$
\\\hline $1$ & $680$ & $293_1,293_2,293_3$
\\ $8$ & $56$ & None
\\ $28$ & $56$ & None
\\ $35_i$ & $56$ & None
\\ $56$ & $56$ & None
\\ \hline\end{tabular}\end{center}


\newpage
\section{\texorpdfstring{${}^3\!D_4(2)$}{3D4(2)}}

\subsection*{Block information}

\begin{center}\begin{tabular}{ccc}
\hline Block & Kernel & Modules
\\\hline $1$ & $1$ & $1,8_1,8_2,8_3,26,48_1,48_2,48_3,160_1,160_2,160_3,246,784_1,784_2,784_3$
\\ $2$ & $1$ & $4096$
\\\hline\end{tabular}\end{center}

\subsection*{Identification of modules}

Choose $8_1$ arbitrarily from the three $8_i$. Define $8_{i+1}$ to lie in the socle of $S^2(8_i)$. As with $\Omega_8^+(2)$, we then define $8_i\otimes 26=48_i\oplus 160_i$, and $748_i$ as the socle of $8_i\otimes 246$.

\subsection*{Extension information}


\begin{center}\begin{tabular}{c|ccccccccccccccc}
Block 1 &$1$&$8_1$&$8_2$&$8_3$&$26$&$48_1$&$48_2$&$48_3$&$160_1$&$160_2$&$160_3$&$246$&$784_1$&$784_2$&$784_3$
\\ \hline $1$&$0$&$0$&$0$&$0$&$2$&$0$&$0$&$0$&$1$&$1$&$1$&$1$&$0$&$0$&$0$
\\$8_1$&$0$&$0$&$0$&$0$&$1$&$1$&$1$&$0$&$0$&$0$&$1$&$0$&$1$&$0$&$0$
\\$8_2$&$0$&$0$&$0$&$0$&$1$&$0$&$1$&$1$&$1$&$0$&$0$&$0$&$0$&$1$&$0$
\\$8_3$&$0$&$0$&$0$&$0$&$1$&$1$&$0$&$1$&$0$&$1$&$0$&$0$&$0$&$0$&$1$
\\$26$&$2$&$1$&$1$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$2$&$0$&$0$&$0$
\\$48_1$&$0$&$1$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\$48_2$&$0$&$1$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\$48_3$&$0$&$0$&$1$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\$160_1$&$1$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&&&&$0$&$0$&$0$&$0$
\\$160_2$&$1$&$0$&$0$&$1$&$0$&$0$&$0$&$0$&&&&$0$&$0$&$0$&$0$
\\$160_3$&$1$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&&&&$0$&$0$&$0$&$0$
\\$246$&$1$&$0$&$0$&$0$&$2$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\$784_1$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\$784_2$&$0$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\$784_3$&$0$&$0$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\end{tabular}\end{center}

\newpage
\section{\texorpdfstring{${}^3\!D_4(3)$}{3D4(3)}}

\subsection*{Block information}

\begin{center}\begin{tabular}{ccc}
\hline Block & Kernel & Modules
\\\hline $1$ & $1$ & $1,8_1,8_2,8_3,28,35_1,35_2,35_3,56_1,56_2,56_3,104_1,104_2,104_3,195,224_1,224_2,224_3,224_4,224_5,224_6$,
\\ &&$322,384_1,384_2,384_3,518_1,518_2,518_3,567_1,567_2,567_3,1176_1,1176_2,1176_3$,
\\ &&$1296_1,1296_2,1296_3,1841,1896_1,1896_2,1896_3,1896_4,1896_5,1896_6,3240_1,3240_2,3240_3$,
\\ &&$3948_1,3948_2,3948_3,4768_1,4768_2,4768_3,7600_1,7600_2,7600_3,8132,8343_1,8343_2,8343_3$,
\\ &&$13776,18144_1,18144_2,18144_3,18144_4,18144_5,18144_6,24240_1,24240_2,24240_3$,
\\ &&$43254_1,43254_2,43154_3,58920_1,58920_2,58920_3,96753,196344_1,196344_2,196344_3$
\\ $2$ & $1$ & $531441$
\\\hline\end{tabular}\end{center}

\subsection*{Extension information}

\begin{center}
\begin{tabular}{ccc}
\hline Module & Tested to & $\Ext^1=k$
\\ \hline $1$ & $567$ & $384$
\\ $8$ & $56$ & None
\\ $28$ & $56$ & None
\\ $35$ & $56$ & None
\\ $56$ & $35$ & None
\\\hline \end{tabular}
\end{center}


\newpage
\section{\texorpdfstring{${}^3\!D_4(5)$}{3D4(5)}}

\subsection*{Block information}
These modules are only necessarily complete up to dimension 10000.
\begin{center}\begin{tabular}{ccc}
\hline Block & Kernel & Modules
\\\hline $1$ & $1$ & $1,8_1,8_2,8_3,28,35_1,35_2,35_3,56_1,56_2,56_3,104_1,104_2,104_3,160_1,160_2,160_3,$
\\ &&$168_1,168_2,168_3,168_4,168_5,168_6,293_1,293_2,293_3,300,350,539_1,539_2,539_3,$
\\ &&$664_1,664_2,664_3,664_4,664_5,664_6,664_7,664_8,664_9,680_1,680_2,680_3,805_1,805_2,805_3,$
\\ &&$904_1,904_2,904_3,1322_1,1322_2,1322_3,1400_1,1400_2,1400_3,1680_1,1680_2,1680_3,1680_4,2451,$
\\ &&$1680_5,1680_6,1925,2360_1,2360_2,2360_3,2400_1,2400_2,2400_3,2400_4,2400_5,2400_6,$
\\ &&$2501,2800_1,2800_2,2800_3,2800_4,2800_5,2800_6,3096_1,3096_2,3096_3,3403_1,3403_2,3403_3,$
\\ &&$3632_1,3632_2,3632_3,3632_4,3632_5,3632_6,3675_1,3675_2,3675_3,4992_1,4992_2,4992_3,$
\\ &&$5480_1,5480_2,5480_3,5775_1,5775_2,5775_3,5775_4,5775_5,5775_6,6600_1,6600_2,6600_3,$
\\ &&$6944_1,6944_2,6944_3,6944_4,6944_5,6944_6,7372_1,7372_2,7372_3,7552_1,7552_2,7552_3,$
\\ &&$8105_1,8105_2,8105_3,8800_1,8800_2,8800_3,9416_1,9416_2,9416_3,$... 
\\ $2$ & $1$ & $244140625$
\\\hline\end{tabular}\end{center}

\subsection*{Extension information}
\begin{center}\begin{tabular}{ccc}
\hline
Module & Tested to & $\Ext^1=k$
\\\hline $1$ & $168$ & None
\\$8_i$ & $168$ & $104_i$
\\$28$ & $56$ & None
\\$35$ & $56$ & None
\\$56$ & $56$ & None
\\ \hline\end{tabular}\end{center}

To distinguish $8_i$ and $104_i$, note that $104_i$ is a composition factor of $S^3(8_i)$, and in fact $S^3(8_i)=8_i/104_i/8_i$.


\newpage
\section{\texorpdfstring{$F_4(2)$}{F4(2)}}

\subsection*{Block information}

\begin{center}\begin{tabular}{ccc}
\hline Block & Kernel & Modules
\\ \hline 1 & $1$ & $1,26_1,26_2,246_1,246_2,676,4096_1,4096_2,6396_1,6936_2,60516$,
\\ && $106496_1,106496_2,1007616_1,1007616_2$
\\ 2 & $1$ & $16777216$
\\\hline\end{tabular}\end{center}


\subsection*{Identification of modules}

Define $26_1$ arbitrarily from the $26$-dimensionals. We define $246_i$ to lie in $\Lambda^2(26_i)$. The module $676=26_1\otimes 26_2$. We have that $4096_i$ lies in $26_i\otimes 246_i$. Then $6376_i=26_i\otimes 246_{3-i}$.

\subsection*{Highest weight labels}

\begin{center}\begin{tabular}{ccccccccc}
\hline $\lambda$&0000&0001&1000&0010&0100&1001&0011&1100
\\\hline $L(\lambda)$ &$1$&$26_1$&$26_2$&$246_1$&$246_2$&$676$&$4096_1$&$4096_2$
\\ $\dim(W(\lambda))$&$1$&$26$&$52$&$273$&$1274$&$1053$&$4096$&$29172$
\\\hline $\lambda$&0101&1010&0110&1011&1101&0111&1110&1111
\\\hline $L(\lambda)$&$6396_1$&$6396_2$&$60516$&$106496_1$&$106496_2$&$1007616_1$&$1007616_2$&$16777216$
\\ $\dim(W(\lambda))$&$19278$&$8424$&$107406$&$106496$&$379848$&$1118208$&$1801371$&$16777216$
\\ \hline
\end{tabular}\end{center}

\subsection*{Extension information}

\begin{center}\begin{tabular}{c|cccccccccc}
Block 1 &$1$&$26_1$&$26_2$&$246_1$&$246_2$&$676$&$4096_1$&$4096_2$&$6396_1$&$6396_2$
\\\hline $1$&$0$&$0$&$0$&$1$&$1$&$2$&$0$&$0$&$0$&$0$
\\ $26_1$&$0$&$0$&$2$&$2$&$0$
\\ $26_2$&$0$&$2$&$0$&$0$&$2$
\end{tabular}\end{center}


\newpage
\section{\texorpdfstring{$G_2(2)'$}{G2(2)'}}

\subsection*{Block information}

\begin{center}\begin{tabular}{ccc}
\hline Block & Kernel & Modules
\\\hline 1 & $1$ & $1$, $6$, $14$
\\ 2 & $1$ & $32$
\\ 3 & $1$ & Dual of block 2
\\\hline\end{tabular}\end{center}

\subsection*{Highest weight labels}

\begin{center}\begin{tabular}{ccccc}
\hline $\lambda$&00&10&01&11
\\ \hline$L(\lambda)$ & $1$&$6$&$14$&$32_1\oplus 32_2$
\\ $\dim(W(\lambda))$ & $1$&$7$&$14$& $64$
\\ \hline
\end{tabular}\end{center}

\subsection*{Extension information}

\begin{center}\begin{tabular}{c|ccc}
Block 1&$1$&$6$&$14$
\\\hline $1$&$0$&$1$&$0$
\\ $6$&$1$&$0$&$1$
\\ $14$&$0$&$1$&$0$
\end{tabular}\end{center}

\subsection*{Socle layers of projective modules}

\[\begin{array}{c}
1
\\6
\\1\oplus14
\\6\oplus6
\\1\oplus1\oplus14
\\6\oplus6
\\1\oplus1\oplus14
\\6\oplus6
\\1\oplus1\oplus14
\\6\oplus6
\\1\oplus1\oplus14
\\6\oplus6
\\1\oplus1\oplus14
\\6\oplus6
\\1\oplus1\oplus14
\\6\oplus6
\\1\oplus14
\\6
\\1\end{array}\qquad
\begin{array}{c}6
\\1\oplus14
\\6\oplus6
\\1\oplus1\oplus14
\\6\oplus6
\\1\oplus1\oplus14
\\6\oplus6
\\1\oplus1\oplus14
\\6\oplus6
\\1\oplus1\oplus14
\\6\oplus6
\\1\oplus1\oplus14
\\6\oplus6
\\1\oplus1\oplus14
\\6\oplus6\oplus6
\\1\oplus1\oplus14
\\6\oplus6
\\1\oplus14
\\6
\end{array}\qquad
\begin{array}{c}14
\\6
\\1
\\6
\\1\oplus14
\\6
\\1
\\6
\\1\oplus14
\\6
\\1
\\6
\\1\oplus14
\\6
\\1\oplus14
\\6
\\1\oplus14
\\6
\\14\end{array}\]


\subsection*{Symmetric and exterior squares of modules}

\begin{center}\begin{tabular}{c|cc}
$M$&$6$&$14$
\\\hline $S^2(M)$&$1,14/6$&$1\oplus (14/6/1/6/1,14/6/1/6/1,14/6/14,14)$
\\ $\Lambda^2(M)$&$1\oplus 14$&$1\oplus (14/6/1/6/1,14/6/1/6/1,14/6/14)$
\end{tabular}\end{center}


\subsection*{Actions of unipotent elements on modules}

\begin{center}\begin{tabular}{c|ccc}
Block 1&$1$&$6$&$14$
\\\hline 2A&$1$&$2^2,1^2$&$2^6,1^2$
\\ 4A[2],4C&$1$&$3^2$&$4^2,3^2$
\\ 8A[A][2]&$1$&$6$&$8,6$
\end{tabular}\end{center}

\newpage
\section{\texorpdfstring{$G_2(3)$}{G2(3)}}

\subsection*{Block information}

\begin{center}\begin{tabular}{ccc}
\hline Block & Kernel & Modules
\\\hline 1 & $1$ & $1$, $7_1$, $7_2$, $27_1$, $27_2$, $49$, $189_1$, $189_2$
\\ 2 & $1$ & $729$
\\\hline\end{tabular}\end{center}

\subsection*{Highest weight labels}

\begin{center}\begin{tabular}{cccccccccc}
\hline $\lambda$&00&10&01&20&02&11&21&12&22
\\ \hline$L(\lambda)$ & $1$&$7_1$&$7_2$&$27_1$&$27_2$&$49$&$189_1$&$189_2$&$729$
\\ $\dim(W(\lambda))$ & $1$&$7$&$14$&$27$&$77$&$64$&$189$&$286$&$729$
\\ \hline
\end{tabular}\end{center}

\subsection*{Extension information}

\begin{center}\begin{tabular}{c|cccccccc}
Block 1 & $1$&$7_1$&$7_2$&$27_1$&$27_2$&$49$&$189_1$&$189_2$
\\\hline $1$&$0$&$0$&$0$&$0$&$0$&$2$&$0$&$0$
\\ $7_1$&$0$&$0$&$2$&$0$&$0$&$1$&$1$&$0$
\\ $7_2$&$0$&$2$&$0$&$0$&$0$&$1$&$0$&$1$
\\ $27_1$&$0$&$0$&$0$&$0$&$0$&$1$&$1$&$0$
\\ $27_2$&$0$&$0$&$0$&$0$&$0$&$1$&$0$&$1$
\\ $49$&$2$&$1$&$1$&$1$&$1$&$0$&$0$&$0$
\\ $189_1$&$0$&$1$&$0$&$1$&$0$&$0$&$0$&$0$
\\ $189_2$&$0$&$0$&$1$&$0$&$1$&$0$&$0$&$0$
\end{tabular}\end{center}

\subsection*{Symmetric and exterior squares of modules}

\begin{center}\begin{tabular}{c|cc}
Block 1 & $7_1$&$7_2$
\\\hline$S^2(M)$&$1\oplus 27_1$&$1\oplus 27_2$
\\ $\Lambda^2(M)$&$7_1/7_2/7_1$&$7_2/7_1/7_2$
\end{tabular}\end{center}

\subsection*{Actions of unipotent elements on modules}


\begin{center}\begin{tabular}{c|cccccccc}
Block 1 & $1$&$7_1$&$7_2$&$27_1$&$27_2$&$49$&$189_1$&$189_2$
\\\hline 3A&$1$&$2^2,1^3$&$3,2^2$&$3^3,2^6,1^6$&$3^9$&$3^{11},2^6,1^4$&$3^{51},2^{12},1^{12}$&$3^{63}$
\\ 3B&$1$&$3,2^2$&$2^2,1^3$&$3^9$&$3^3,2^6,1^6$&$3^{11},2^6,1^4$&$3^{63}$&$3^{51},2^{12},1^{12}$
\\ 3C&$1$&$3,2^2$&$3,2^2$&$3^9$&$3^9$&$3^{15},1^4$&$3^{63}$&$3^{63}$
\\3D, 3E&$1$&$3^2,1$&$3^2,1$&$3^9$&$3^9$&$3^{16},1$&$3^{63}$&$3^{63}$
\\ 9A[C],9B[2][C]&$1$&$7$&$7$&$9^3$&$9^3$&$9^5,3,1$&$9^{21}$&$9^{21}$
\end{tabular}\end{center}


\newpage
\section{\texorpdfstring{$G_2(4)$}{G2(4)}}

\subsection*{Block information}

\begin{center}\begin{tabular}{ccc}
\hline Block & Kernel & Modules
\\\hline $1$ & $1$ & $1$, $6_1$, $6_2$, $14_1$, $14_2$, $36$, $64_1$, $64_2$, $84_{1,2}$, $84_{2,1}$, $196$, $384_{1,2}$, $384_{2,1}$, $896_{1,2}$, $896_{2,1}$
\\ $2$ & $1$ & $4096$
\\\hline\end{tabular}\end{center}

\subsection*{Identification of modules}


\begin{enumerate}
\item $84_{1,2}=6_1\otimes 14_2$, $84_{2,1}=14_1\otimes 6_2$
\item $384_{1,2}=6_1\otimes 64_2$, $384_{2,1}=64_1\otimes 6_2$
\item $896_{1,2}=14_1\otimes 64_2$, $896_{2,1}=64_1\otimes 14_2$
\end{enumerate}

\subsection*{Extension information}

\begin{center}\begin{tabular}{c|ccccccccccccccc}
Block 1&$1$&$6_1$&$6_2$&$14_1$&$14_2$&$36$&$64_1$&$64_2$&$84_{1,2}$&$84_{2,1}$&$196$&$384_{1,2}$&$384_2$&$896_{1,2}$&$896_2$
\\ \hline $1$&$0$&$1$&$1$&$0$&$0$&$1$&$0$&$0$&$1$&$1$&$0$&$0$&$0$&$0$&$0$
\\ $6_1$&$1$&$0$&$1$&$0$&$1$&$1$&$0$&$0$&$0$&$1$&$1$&$0$&$0$&$0$&$0$
\\ $6_2$&$1$&$1$&$0$&$1$&$0$&$1$&$0$&$0$&$1$&$0$&$1$&$0$&$0$&$0$&$0$
\\ $14_1$&$0$&$0$&$1$&$0$&$0$&$1$&$0$&$0$&$1$&$1$&$0$&$0$&$1$&$0$&$1$
\\ $14_2$&$0$&$1$&$0$&$0$&$0$&$1$&$0$&$0$&$1$&$1$&$0$&$1$&$0$&$1$&$0$
\\ $36$&$1$&$1$&$1$&$1$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\ $64_1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$1$&$1$&$0$&$2$&$0$&$1$
\\ $64_2$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$1$&$0$&$1$&$2$&$0$&$1$&$0$
\\ $84_{1,2}$&$1$&$0$&$1$&$1$&$1$&$0$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\ $84_{2,1}$&$1$&$1$&$0$&$1$&$1$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\ $196$&$0$&$1$&$1$&$0$&$0$&$0$&$1$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\ $384_{1,2}$&$0$&$0$&$0$&$0$&$1$&$0$&$0$&$2$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\ $384_{2,1}$&$0$&$0$&$0$&$1$&$0$&$0$&$2$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\ $896_{1,2}$&$0$&$0$&$0$&$0$&$1$&$0$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\ $896_{2,1}$&$0$&$0$&$0$&$1$&$0$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\end{tabular}\end{center}


\subsection*{Symmetric and exterior squares of modules}

\begin{center}\begin{tabular}{c|cc}
Block 1&$6_1$&$14_1$
\\ \hline $S^2(M)$&$1,14_1/6_2$&$1\oplus (14_1/6_2/1/6_1/36/6_1/1/6_2/14_1,14_2)$
\\ $\Lambda^2(M)$&$1\oplus 14_1$&$1\oplus (14_1/6_2/1/6_1/36/6_1/1/6_2/14_1)$
\end{tabular}\end{center}

\subsection*{Actions of unipotent elements on modules}

\begin{center}\begin{tabular}{c|ccccccccc}
Block 1&$1$&$6_i$&$14_i$&$36$&$64_i$&$84_{i,j}$&$196$&$384_{i,j}$&$896_{i,j}$
\\\hline 2A& $1$&$2^2,1^2$&$2^6,1^2$&$2^{16},1^4$&$2^{32}$&$2^{40},1^4$&$2^{96},1^4$&$2^{192}$&$2^{448}$
\\2B&$1$&$2^3$&$2^6,1^2$&$2^{18}$&$2^{32}$&$2^{42}$&$2^{96},1^4$&$2^{192}$&$2^{448}$
\\4A[A],4B[A],4C[A]&$1$&$3^2$&$4^2,3^2$&$4^8,1^4$&$4^{16}$&$4^{20},1^4$&$4^{48},1^4$&$4^{96}$&$4^{224}$
\\8A[A],8B[C]&$1$&$6$&$8,6$&$8^4,2^2$&$8^8$&$8^{10},2^2$&$8^{24},2^2$&$8^{48}$&$8^{112}$
\end{tabular}\end{center}


\newpage
\section{\texorpdfstring{$G_2(5)$}{G2(5)}}

\subsection*{Block information}

\begin{center}\begin{tabular}{ccc}
\hline Block & Kernel & Modules
\\\hline 1 & $1$ & $1$, $7$, $14$, $27$, $64$, $77_1$, $77_2$, $97$, $182$, $189$, $196$, $371$, $469$, $483$,
\\ && $721$, $792$, $1344$, $1715$, $2008$, $2247$, $2380$, $2667$, $4830$, $8456$
\\ 2 & $1$ & $15625$
\\\hline\end{tabular}\end{center}

\subsection*{Highest weight labels}

\begin{center}\begin{tabular}{cccccccccccccc}
\hline $\lambda$&00&10&01&20&11&30&02&12&40&21&03&31&41
\\\hline $L(\lambda)$ &$1$&$7$&$14$&$27$&$64$&$77_1$&$77_2$&$97$&$182$&$189$&$196$&$371$&$469$
\\ $\dim(W(\lambda))$&$1$&$7$&$14$&$27$&$64$&$77$&$77$&$286$&$182$&$189$&$273$&$448$&$924$
\\\hline $\lambda$&22&04&13&32&33&23&14&42&24&43&34&44&
\\\hline $L(\lambda)$&$483$&$721$& $792$&$1344$&$1715$&$2008$&$2247$&$2380$&$2667$&$4830$&$8456$&$15625$&
\\ $\dim(W(\lambda))$&$729$&$748$&$896$&$1547$&$4096$&$2079$&$2261$&$2926$&$4914$&$7293$&$9177$&$15625$&
\\\hline
\end{tabular}\end{center}


\subsection*{Extension information}

There are no extensions between $8456$ and modules of dimension greater than $792$.

\begin{center}\begin{tabular}{c|ccccccccccccccccc}
Block 1& $1$&$7$&$14$&$27$&$64$&$77_1$&$77_2$&$97$&$182$&$189$&$196$&$371$&$469$&$483$&$721$&$792$&$1344$
\\\hline $1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$1$&$0$&$0$&$0$
\\ $7$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$1$&$0$&$0$&$1$&$1$&$1$&$1$&$0$&$1$&$1$
\\ $14$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$1$&$0$&$0$&$2$

\\ $27$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$1$&$0$&$0$&$1$
\\ $64$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$1$&$0$
\\ $77_1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\ $77_2$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\ $97$&$0$&$1$&$0$&$1$&$1$&$0$&$0$&$0$
\\ $182$&$0$&$0$&$0$&$0$&$0$
\\ $189$&$0$&$0$&$0$&$0$
\\ $196$&$0$&$1$&$1$&$1$
\\ $371$&$0$&$1$&$0$
\\ $469$&$0$&$1$&$0$
\\ $483$&$1$&$1$&$2$
\\ $721$&$0$&$0$
\\ $792$&$0$&$1$
\\ $1344$&$0$&$1$
\\ $1715$&$1$
\\ $2008$&$0$
\\ $2247$&$0$
\\ $2380$&$0$
\\ $2667$&$0$
\\ $4830$&$0$
\\ $8456$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$1$&$1$&$1$&$0$
\end{tabular}\end{center}

\newpage
\section{\texorpdfstring{$G_2(7)$}{G2(7)}}

\subsection*{Block information}

\begin{center}\begin{tabular}{ccc}
\hline Block & Kernel & Modules
\\\hline 1 & $1$ & $1$, $7$, $14$, $26$, $38$, $77_1$, $77_2$, $182$, $189$, $248$, $267$, $273$, $378$, $448$, $481$, $532$, $714$, $896$, $924$,
\\ && $1247$, $1337$, $1351$, $1547$, $1827$, $1853$, $2030$, $3003$, $3360$, $3419$, $3752$, $4368$, $5397$, $6578$,
\\ && $8064$, $9471$, $9681$, $10472$, $11413$, $12726$, $17374$, $19201$, $24661$, $26902$, $44128$, $56266$, $77721$
\\ 2 & $1$ & $114649$
\\\hline\end{tabular}\end{center}

\subsection*{Highest weight labels}

\begin{center}\begin{tabular}{ccccccccccc}
\hline $\lambda$&00&10&01&20&11&30&02&40&21&12
\\\hline $L(\lambda)$ &$1$&$7$&$14$&$26$&$38$&$77_1$&$77_2$&$182$&$189$&$248$
\\ $\dim(W(\lambda))$&$1$&$7$&$14$&$27$&$64$&$77$&$77$&$182$&$189$&$286$
\\\hline $\lambda$&04&03&50&31&22&23&60&13&41&51
\\\hline $L(\lambda)$&$267$&$273$&$378$&$448$&$481$&$532$&$714$&$896$&$924$&$1247$
\\ $\dim(W(\lambda))$&$748$&$273$&$378$&$448$&$729$&$2079$&$714$&$896$&$924$&$1728$
\\\hline $\lambda$&14&05&32&52&33&42&61&06&43&24
\\\hline $L(\lambda)$&$1337$&$1351$&$1547$&$1827$&$1853$&$2030$&$3003$&$3360$&$3419$&$3752$
\\ $\dim(W(\lambda))$&$2261$&$1729$&$1547$&$5103$&$4096$&$2926$&$3003$&$3542$&$7293$&$4914$
\\\hline $\lambda$&15&62&35&34&16&25&53&64&44&63
\\\hline $L(\lambda)$&$4368$&$5397$&$6578$&$8064$&$9471$&$9681$&$10472$&$11375$&$11413$&$12726$
\\ $\dim(W(\lambda))$&$4928$&$8372$&$18304$&$9177$&$9660$&$10206$&$12096$&$37961$&$15625$&$19019$
\\\hline $\lambda$&36&45&26&54&55&46&65&56&66&
\\\hline $L(\lambda)$&$14294$&$17374$&$19201$&$24661$&$26902$&$44128$&$56266$&$77721$&$114649$&
\\ $\dim(W(\lambda))$&$33495$&$30107$&$19278$&$24948$&$46656$&$53599$&$69160$&$81081$&$114649$&
\\\hline
\end{tabular}\end{center}




\subsection*{Extension information}


\begin{center}\begin{tabular}{ccc}\hline
Module & Tested to & $\Ext^1=k$
\\ \hline $1$&$4368$&$26$, $2030$
\\ $7$&$714$ & None
\\ $14$&$248$ & None
\\ $26$&$189$&$1$, $38$
\\ $38$&$189$&$26$
\\ $77_1$ & $77$ & None
\\ $77_2$ & $77$ & None
\\ \hline\end{tabular}\end{center}

\subsection*{Actions of unipotent elements on modules}

\begin{center}\begin{tabular}{c|cccccccc}
Block 1&$1$&$7$&$14$&$26$&$38$&$77_1$&$77_2$
\\\hline 7A& $1$&$2^2,1^3$&$3,2^4,1^3$&$3^3,2^6,1^5$&$4^2,3^5,2^6,1^3$&$4^4,3^9,2^{12},1^{10}$&$5,4^4,3^{10},2^{10},1^6$
\\ 7B&$1$&$3,2^2$&$4^2,3,1^3$&$5,4^2,3^3,2^2$&$6^2,5^3,3,2^4$&$7,6^2,5^3,4^6,3^4,2^2,1^3$&$7^3,6^2,5,4^6,3^3,1^6$
\\ 7C, 7D, 7E&$1$&$3^2,1$&$5,3^3$&$5^3,3^3,1^2$&$7^2,5^3,3^3$&$7^4,5^5,3^7,1^3$&$7^5,5^6,3^3,1^3$
\\ 7F&$1$&$7$&$7^2$&$7^3,5$&$7^5,3$&$7^{11}$&$7^{11}$
\end{tabular}\end{center}

\subsection*{Traces of semisimple elements on modules}

\begin{center}\begin{tabular}{c|cccccccccccccccc}
Block 1&$1$&$7$&$14$&$26$&$38$&$77_1$&$77_2$&$182$&$189$&$248$&$267$&$273$&$378$&$448$&$481$
\\\hline 2A&$1$&$-1$&$-2$&$2$&$-2$&$-3$&$5$&$6$&$-3$&$0$&$3$&$-7$&$-6$&$0$&$9$
\\ 3A&$1$&$-2$&$5$&$-1$&$-7$&$5$&$14$&$-7$&$0$&$-13$&$42$&$30$&$0$&$16$&$13$
\\ 3B&$1$&$1$&$-1$&$-1$&$-1$&$2$&$-1$&$2$&$0$&$2$&$0$&$3$&$0$&$-2$&$-2$
\\ 4A[A]&$1$&$3$&$2$&$2$&$-2$&$1$&$-3$&$2$&$-3$&$-4$&$3$&$-3$&$6$&$0$&$1$
\\ 4B[A]&$1$&$-1$&$2$&$-2$&$2$&$1$&$1$&$2$&$-3$&$0$&$7$&$1$&$-2$&$0$&$-3$
\\ 6A[A,A]&$1$&$2$&$1$&$-1$&$1$&$-3$&$2$&$-3$&$0$&$3$&$6$&$2$&$0$&$0$&$-3$
\\ 6B[B,A]&$1$&$5$&$7$&$11$&$7$&$18$&$11$&$18$&$24$&$6$&$-12$&$-1$&$12$&$18$&$-6$
\\ 6C[B,A]&$1$&$-1$&$1$&$-1$&$1$&$0$&$-1$&$0$&$0$&$0$&$0$&$-1$&$0$&$0$&$0$
\end{tabular}\end{center}

\end{document}



\newpage
\section{\texorpdfstring{$G_2(8)$}{G2(8)}}

\subsection*{Block information}

\begin{center}\begin{tabular}{ccc}
\hline Block & Kernel & Modules
\\\hline $1$ & Yes & $1$, $6_1$, $6_2$, $6_3$, $14_1$, $14_2$, $14_3$, $36_1$, $36_2$, $36_3$, $64_1$, $64_2$, $64_3$, $84_{1,2}$, $84_{1,3}$,
\\ && $84_{2,1}$, $84_{2,3}$, $84_{3,1}$, $84_{3,2}$, \dots
\\\hline\end{tabular}\end{center}

\subsection*{Identification of Modules}

Write $36_i=6_{i+1}\otimes 6_{i+2}$, $84_{i,j}=6_i\otimes 14_j$.
\subsection*{Extension information}

\begin{center}\begin{tabular}{c|ccccccccccccc}
Block 1&$1$&$6_1$&$6_2$&$6_3$&$14_1$&$14_2$&$14_3$&$36_{1,2}$&$36_{2,3}$&$36_{1,3}$&$64_1$&$64_2$&$64_3$
\\\hline $1$&$0$&$1$&$1$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\ $6_1$&$1$&$0$&$0$&$0$&$0$&$0$&$1$&$1$&$0$&$1$&$0$&$0$&$0$
\\ $14_1$&$0$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$1$&$0$&$0$&$0$
\\ $36_{1,2}$&$0$&$1$&$1$&$0$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\ $64_1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\ $84_{1,2}$&$0$&$0$&$1$&$0$&$0$&$1$&$0$&$0$&$1$&$0$&$0$&$1$&$0$
\\ $84_{2,1}$&$1$&$0$&$0$&$0$&$1$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\end{tabular}\end{center}


\subsection*{Actions of unipotent elements on modules}

\begin{center}\begin{tabular}{c|cccccc}
Block 1&$1$&$6_i$&$14_i$&$36_i$&$64_i$&$84_{i,j}$
\\ \hline 2A&$1$&$2^2,1^2$&$2^6,1^2$&$2^{16},1^4$&$2^{32}$&$2^{40},1^4$
\\ 2B &$1$&$2^3$&$2^6,1^2$&$2^{18}$&$2^{32}$&$2^{42}$
\\ 4A[A], 4B[A],4C[A]&$1$&$3^2$&$4^2,3^2$&$4^8,1^4$&$4^{16}$&$4^{20},1^4$
\\ 8A[A][2]&$1$&$6$&$8,6$&$8^4,2^2$&$8^8$&$8^{10},2^2$
\end{tabular}\end{center}

\subsection*{Traces of semisimple elements on modules}

\begin{center}\begin{tabular}{c|ccccccc}
Block 1&$1$&$6_1$&$6_2$&$6_3$&$14_1$&$14_2$&$14_3$
\\\hline3A&$1$&$-3$&$-3$&$-3$&$5$&$5$&$5$
\\3B&$1$&$0$&$0$&$0$&$-1$&$-1$&$-1$
\\ 7A[2]&$1$&$2\lambda_1+2$&$2\lambda_2+2$&$2\lambda_3+2$&$4\lambda_1+\lambda_2+4$&$4\lambda_2+\lambda_3+4$&$4\lambda_3+\lambda_1+4$
\\ 7D[2]&$1$&$2\lambda_1+\lambda_2$&$2\lambda_2+\lambda_3$&$2\lambda_3+\lambda_2$&$\lambda_1+\lambda_3+3$&$\lambda_1+\lambda_2+3$&$\lambda_2+\lambda_3+3$
\\7G&$1$&$-1$&$-1$&$-1$&$0$&$0$&$0$
\end{tabular}\end{center}

\begin{center}\begin{tabular}{c|ccc}
Block 1&$36_1$&$36_2$&$36_3$
\\\hline3A&$9$&$9$&$9$
\\3B&$0$&$0$&$0$
\\ 7A[2]&$4\lambda_2$&$4\lambda_3$&$4\lambda_1$
\\ 7D[2]&$5\lambda_1+2\lambda_2+1$&$5\lambda_2+2\lambda_3+1$&$5\lambda_3+2\lambda_1+1$
\\7G&$1$&$1$&$1$
\end{tabular}\end{center}

Here, $\lambda_i=\zeta_7^i+\zeta_7^{-i}$



\newpage
\section{${}^2G_2(3)'=\SL_2(8)$}

ALL CHECKED

\subsection*{Block information}

\begin{center}\begin{tabular}{ccc}
\hline Block & Kernel & Modules
\\ \hline $1$ & $1$ & $1$, $7$
\\ $2$ & $1$ & $9_1$
\\ $3$ & $1$ & $9_2$
\\ $4$ & $1$ & $9_3$
\\\hline\end{tabular}\end{center}

\subsection*{Extension information}

\begin{center}\begin{tabular}{c|cc}
Block 1&$1$&$7$
\\\hline $1$ &$0$&$1$
\\ $7$&$1$&$1$
\end{tabular}\end{center}
This table is for both $\Ext^1$ and $\Ext^2$.

\subsection*{Socle layers of projective modules}

\[\begin{array}{c}
1
\\7
\\1\end{array}\qquad
\begin{array}{c}
7
\\7
\\7
\\1\oplus7
\\7\end{array}\]
\subsection*{Symmetric and exterior squares of modules}

\begin{center}\begin{tabular}{c|ccc}
Block 1,2&$1$&$7$&$9_1$
\\ \hline $S^2(M)$&$1$&$1\oplus 9_1\oplus 9_2\oplus 9_3$&$P(1)\oplus 9_1\oplus 9_2^{\oplus 2}\oplus 9_3$
\\ $\Lambda^2(M)$&$0$&$7/7/7$&$P(7)$
\end{tabular}\end{center}

\subsection*{Actions of unipotent elements on modules}

\begin{center}\begin{tabular}{c|ccc}
Block 1,2,3,4&$1$&$7$&$9_i$
\\ \hline 3A&$1$&$3,2^2$&$3^3$
\\ 9A[3]&$1$&$7$&$9$
\end{tabular}\end{center}

\subsection*{Traces of semisimple elements on modules}

\begin{center}\begin{tabular}{c|ccccc}
Block 1,2,3,4&$1$&$7$&$9_1$&$9_2$&$9_3$
\\ \hline 2A&$1$&$-1$&$1$&$1$&$1$
\\ 7A[3]&$1$&$0$&$\lambda_1$&$\lambda_2$&$\lambda_3$
\end{tabular}\end{center}
Here, $\lambda_i=\zeta_7^i+\zeta_7^{-i}$.


\newpage
\section{$\Alt(10)$, characteristic $3$}

\subsection*{Block information}

\begin{center}\begin{tabular}{ccc}
\hline Block & Defect & Modules
\\ $1$ & $4$ & $1$, $34$, $41$, $84$, $224$
\\ $2$ & $2$ & $9$, $36$, $90$, $126$, $279$
\\\hline\end{tabular}\end{center}

\subsection*{Extension information}


\subsection*{Radical layers of projectives}

\[ 
\begin{array}{c}
1 
\\34\oplus41\oplus84\oplus224
\\1\oplus1\oplus1\oplus1\oplus34\oplus41\oplus41\oplus84\oplus224\oplus224
\\1\oplus1\oplus1\oplus34\oplus34\oplus34\oplus41\oplus41\oplus41\oplus84\oplus84\oplus224\oplus224
\\1\oplus1\oplus1\oplus1\oplus34\oplus41\oplus41\oplus84\oplus224\oplus224
\\34\oplus41\oplus84\oplus224
\\1\end{array}\qquad \begin{array}{c}
34
\\1\oplus41\oplus224
\\1\oplus34\oplus34\oplus41\oplus84\oplus224
\\1\oplus1\oplus1\oplus34\oplus41\oplus41\oplus84\oplus224
\\1\oplus34\oplus34\oplus41\oplus84\oplus224
\\1\oplus41\oplus224
\\34
\end{array}\]

\[ 
\begin{array}{c}
41
\\1\oplus34\oplus84\oplus224
\\1\oplus1\oplus34\oplus41\oplus41\oplus41\oplus84\oplus224
\\1\oplus1\oplus1\oplus34\oplus34\oplus41\oplus84\oplus84\oplus224\oplus224
\\1\oplus1\oplus34\oplus41\oplus41\oplus41\oplus84\oplus224
\\1\oplus34\oplus84\oplus224
\\41
\end{array}\qquad
\begin{array}{c}
84
\\1\oplus41\oplus84
\\1\oplus34\oplus41\oplus84\oplus84\oplus224
\\1\oplus1\oplus34\oplus41\oplus41\oplus84\oplus224
\\1\oplus34\oplus41\oplus84\oplus84\oplus224
\\1\oplus41\oplus84
\\84
\end{array}\]
\[\begin{array}{c}
224
\\1\oplus34\oplus41\oplus224
\\1\oplus1\oplus34\oplus41\oplus84\oplus224\oplus224
\\1\oplus1\oplus34\oplus41\oplus41\oplus84\oplus224\oplus224
\\1\oplus1\oplus34\oplus41\oplus84\oplus224\oplus224
\\1\oplus34\oplus41\oplus224
\\224
\end{array}\]

\[\begin{array}{c}9
\\126\oplus279
\\9\oplus9\oplus36\oplus90
\\126\oplus279
\\9\end{array}\quad
\begin{array}{c}36
\\126\oplus279
\\9\oplus36\oplus36\oplus90
\\126\oplus279
\\36\end{array}\quad
\begin{array}{c}90
\\126
\\9\oplus36\oplus90
\\126
\\90\end{array}\quad
\begin{array}{c}126
\\9\oplus36\oplus90
\\126\oplus126\oplus279
\\9\oplus36\oplus90
\\126\end{array}\quad
\begin{array}{c}279
\\9\oplus36
\\126\oplus279
\\9\oplus36
\\279
\end{array}\]



\newpage
\section{$\Alt(11)$, characteristic $3$}

\subsection*{Block information}

\begin{center}\begin{tabular}{ccc}
\hline Block & Defect & Modules
\\\hline $1$ & $4$ & $1$, $10$, $34$, $109$, $120$, $131$, $210$, $320$, $714$, $791$
\\ $2$ & $2$ & $45$, $126_1$, $126_2$, $693$
\\ $3$ & $1$ & $594$
\\\hline\end{tabular}\end{center}

\subsection*{Socle layers of projectives}
\[ 
\begin{array}{c}
\\1
\\109\;\;120\;\;714
\\1\;\;1\;\;1
\\109\;\;109\;\;120\;\;714
\\1\;\;1\;\;1\;\;1\;\;34\;\;210\;\;320
\\109\;\;109\;\;120\;\;120\;\;131\;\;714\;\;791
\\1\;\;1\;\;1\;\;1\;\;1\;\;34\;\;34\;\;210\;\;210\;\;320
\\10\;\;109\;\;109\;\;120\;\;120\;\;131\;\;714\;\;714\;\;791
\\1\;\;1\;\;1\;\;1\;\;1\;\;34\;\;210\;\;320
\\10\;\;10\;\;109\;\;109\;\;120\;\;120\;\;131\;\;131\;\;714\;\;714\;\;791
\\1\;\;1\;\;1\;\;1\;\;1\;\;34\;\;34\;\;210\;\;210\;\;320
\\109\;\;120\;\;131\;\;714\;\;791
\\1\end{array}
\quad
\begin{array}{c}10
\\34\;\;210\;\;320
\\120\;\;714
\\1
\\10\;\;10\;\;10\;\;109\;\;131\;\;791
\\1\;\;1\;\;34\;\;34\;\;34\;\;210\;\;210\;\;320\;\;320
\\10\;\;10\;\;10\;\;120\;\;131\;\;714\;\;791
\\34\;\;210\;\;320
\\10
\end{array}\]


\[\begin{array}{c}
34
\\714
\\1
\\10\;\;109\;\;131\;\;791
\\1\;\;1\;\;34\;\;210\;\;320
\\120\;\;714
\\1\;\;34\;\;34\;\;210
\\10\;\;10\;\;10\;\;109\;\;120\;\;131\;\;131\;\;714\;\;791
\\1\;\;1\;\;34\;\;34\;\;34\;\;210\;\;210\;\;320
\\10\;\;131\;\;714\;\;791
\\34
\end{array}\quad
\begin{array}{c}
109
\\1
\\120\;\;714
\\1\;\;1
\\109\;\;109
\\1\;\;1\;\;34\;\;210\;\;320
\\120\;\;120\;\;714
\\1\;\;1
\\10\;\;109\;\;109\;\;131\;\;791
\\1\;\;1\;\;34\;\;210\;\;320
\\120\;\;714
\\1
\\109
\end{array}
\quad
\begin{array}{c}
120
\\1
\\109
\\1\;\;210\;\;320
\\120\;\;120\;\;714
\\1\;\;1
\\10\;\;109\;\;109\;\;131\;\;791
\\1\;\;1\;\;34\;\;210\;\;320
\\120\;\;120\;\;714
\\1\;\;1\;\;34\;\;210\;\;320
\\10\;\;109\;\;120\;\;120\;\;131\;\;791
\\1\;\;210\;\;320
\\120
\end{array}\]
\[\begin{array}{c}
131
\\1\;\;34\;\;210
\\120\;\;714
\\1
\\10\;\;109\;\;131\;\;131
\\1\;\;1\;\;34\;\;34\;\;210\;\;210
\\10\;\;120\;\;131\;\;131\;\;714
\\1\;\;34\;\;210
\\131\end{array}
\quad
\begin{array}{c}
210
\\120\;\;714
\\1
\\10\;\;109\;\;131
\\1\;\;1\;\;34\;\;210\;\;320
\\120\;\;714
\\1\;\;34\;\;210\;\;210
\\10\;\;10\;\;109\;\;120\;\;131\;\;131\;\;714\;\;791
\\1\;\;1\;\;34\;\;34\;\;210\;\;210\;\;210\;\;320
\\10\;\;120\;\;131\;\;714
\\210\end{array}
\quad
\begin{array}{c}
320
\\120
\\1
\\10\;\;109\;\;791
\\1\;\;34\;\;210\;\;320
\\120\;\;714
\\1\;\;320
\\10\;\;10\;\;109\;\;120\;\;791
\\1\;\;34\;\;210\;\;320\;\;320
\\10\;\;120\;\;791
\\320\end{array}\]


\[\begin{array}{c}
714
\\1
\\109
\\1\;\;34\;\;210
\\120\;\;714
\\1
\\10\;\;109\;\;131\;\;791
\\1\;\;1\;\;34\;\;210\;\;320
\\120\;\;714\;\;714
\\1\;\;1\;\;34\;\;210
\\10\;\;109\;\;131\;\;714\;\;791
\\1\;\;34\;\;210
\\714
\end{array}\quad
\begin{array}{c}
791
\\1\;\;34\;\;320
\\120\;\;714
\\1
\\10\;\;109\;\;791
\\1\;\;34\;\;210\;\;320
\\10\;\;120\;\;714\;\;791
\\1\;\;34\;\;320
\\791
\end{array}\]

\[
\begin{array}{c}
45
\\126_1\;\;126_2
\\45\;\;693\;\;693
\\126_1\;\;126_2
\\45
\end{array}\quad
\begin{array}{c}
126_1
\\45\;\;693
\\126_1\;\;126_2\;\;126_2
\\45\;\;693
\\126_1
\end{array}\quad
\begin{array}{c}
126_2
\\45\;\;693
\\126_1\;\;126_1\;\;126_2
\\45\;\;693
\\126_2
\end{array}\quad
\begin{array}{c}
693
\\126_1\;\;126_2
\\45\;\;45\;\;693
\\126_1\;\;126_2
\\693
\end{array}\]

\newpage
\section{$\Alt(12)$, characteristic $3$}

\subsection*{Block information}

\begin{center}\begin{tabular}{ccc}
\hline Block & Defect & Modules
\\\hline $1$ & $5$ & $1$, $10$, $45$, $120$, $126$, $126^*$, $131$, $143$, $210$, $714$, $714^*$, $1013$, $1936$
\\ $2$ & $2$ & $54$, $297$, $945$, $1431$, $1728$
\\ $3$ & $1$ & $891$, $3564$
\\ $4$ & $0$ & $2673$
\\\hline\end{tabular}\end{center}

\subsection*{Extension information}

\begin{center}\begin{tabular}{c|ccccccccccccc}
Block 1& $1$&$10$&$45$&$120$&$126$&$126^*$&$131$&$143$&$210$&$714$&$714^*$&$1013$&$1936$
\\\hline $1$&$0$&$1$&$0$&$1$&$0$&$0$&$1$&$1$&$0$&$1$&$1$&$0$&$1$
\\$10$&$1$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$1$&$0$&$0$&$1$&$1$
\\$45$&$0$&$1$&$0$&$1$&$1$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\\$120$&$1$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$2$&$0$&$0$&$1$&$0$
\\$126$&$0$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$1$&$0$&$0$&$1$&$0$
\\$126^*$&$0$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$1$&$0$&$0$&$1$&$0$
\\$131$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$1$&$1$&$0$&$0$&$1$&$0$
\\$143$&$1$&$0$&$0$&$0$&$0$&$0$&$1$&$0$&$0$&$0$&$0$&$0$&$0$
\\$210$&$0$&$1$&$0$&$2$&$1$&$1$&$1$&$0$&$0$&$1$&$1$&$0$&$0$
\\$714$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$1$&$0$&$0$&$0$&$0$
\\$714^*$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$1$&$0$&$0$&$0$&$0$
\\$1013$&$0$&$1$&$0$&$1$&$1$&$1$&$1$&$0$&$0$&$0$&$0$&$0$&$0$
\\$1936$&$1$&$1$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$&$0$
\end{tabular}\end{center}



\subsection*{Socle layers of projectives}

\[\begin{array}{c}
10
\\1\;\;1936
\\1\;\;45\;\;143\;\;210
\\1\;\;10\;\;10\;\;120\;\;120\;\;714\;\;714^*\;\;1013
\\1\;\;1\;\;1\;\;10\;\;120\;\;120\;\;131\;\;714\;\;714^*
\\1\;\;1\;\;1\;\;10\;\;126\;\;126^*\;\;131\;\;143\;\;143\;\;143\;\;1936
\\1\;\;1\;\;1\;\;1\;\;45\;\;126\;\;126^*\;\;143\;\;143\;\;210\;\;210\;\;210\;\;1013\;\;1013\;\;1936\;\;1936
\\1\;\;1\;\;1\;\;10\;\;10\;\;45\;\;45\;\;120\;\;120\;\;120\;\;143\;\;210\;\;210\;\;210\;\;714\;\;714\;\;714^*\;\;714^*\;\;1013
\\1\;\;1\;\;1\;\;1\;\;1\;\;10\;\;10\;\;10\;\;10\;\;120\;\;120\;\;120\;\;131\;\;131\;\;714\;\;714\;\;714^*\;\;714^*
\\1\;\;1\;\;1\;\;10\;\;10\;\;120\;\;126\;\;126^*\;\;131\;\;143\;\;143\;\;143\;\;1936\;\;1936
\\1\;\;1\;\;1\;\;1\;\;1\;\;1\;\;45\;\;45\;\;126\;\;126^*\;\;143\;\;143\;\;210\;\;210\;\;210\;\;1013\;\;1936
\\1\;\;1\;\;10\;\;10\;\;10\;\;10\;\;10\;\;120\;\;120\;\;120\;\;126\;\;126\;\;126^*\;\;126^*\;\;131\;\;131\;\;143\;\;210\;\;210\;\;714\;\;714^*\;\;1013
\\1\;\;1\;\;1\;\;1\;\;45\;\;45\;\;45\;\;120\;\;210\;\;210\;\;210\;\;714\;\;714^*\;\;1013\;\;1013\;\;1013
\\1\;\;10\;\;10\;\;10\;\;10\;\;120\;\;120\;\;126\;\;126\;\;126^*\;\;126^*\;\;131\;\;131\;\;143
\\1\;\;45\;\;210\;\;1013\;\;1936
\\10\end{array}\]

\[\begin{array}{c}
45
\\10\;\;120
\\1
\\126\;\;126^*\;\;143
\\1\;\;45\;\;210\;\;210\;\;1013\;\;1013
\\10\;\;120\;\;120\;\;120\;\;714\;\;714^*
\\1\;\;1\;\;1\;\;10\;\;10\;\;131\;\;131
\\126\;\;126^*\;\;143\;\;143\;\;143\;\;1936\;\;1936
\\1\;\;1\;\;1\;\;45\;\;45\;\;210\;\;210
\\10\;\;10\;\;120\;\;120\;\;126\;\;126^*\;\;714\;\;714^*
\\1\;\;1\;\;1\;\;45\;\;45\;\;210\;\;210\;\;1013\;\;1013
\\10\;\;10\;\;10\;\;120\;\;120\;\;120\;\;126\;\;126\;\;126^*\;\;126^*\;\;131\;\;131\;\;143
\\1\;\;45\;\;45\;\;210\;\;210\;\;1013\;\;1013
\\10\;\;120\;\;126\;\;126^*
\\45\end{array}\]

\[\begin{array}{c}
120
\\1 
\\143\;\;1936 
\\1\;\;1\;\;45\;\;210\;\;210\;\;1013 
\\10\;\;10\;\;120\;\;120\;\;120\;\;120\;\;120\;\;714\;\;714\;\;714^*\;\;714^*
\\1\;\;1\;\;1\;\;1\;\;1\;\;1\;\;10\;\;10\;\;131\;\;131\;\;131 
\\126\;\;126\;\;126^*\;\;126^*\;\;143\;\;143\;\;143\;\;143\;\;143\;\;1936\;\;1936\;\;1936 
\\1\;\;1\;\;1\;\;1\;\;1\;\;1\;\;1\;\;1\;\;45\;\;45\;\;45\;\;210\;\;210\;\;210\;\;210\;\;210\;\;210\;\;210\;\;1013,
1013\;\;1013 
\\10\;\;10\;\;10\;\;120\;\;120\;\;120\;\;120\;\;120\;\;120\;\;120\;\;143\;\;714\;\;714\;\;714\;\;714\;\;714^*\;\;714^*\;\;714^*\;\;714^*
\\1\;\;1\;\;1\;\;1\;\;1\;\;1\;\;1\;\;1\;\;10\;\;10\;\;10\;\;131\;\;131\;\;131\;\;131 
\\10\;\;120\;\;120\;\;126\;\;126\;\;126^*\;\;126^*\;\;143\;\;143\;\;143\;\;143\;\;143\;\;1936\;\;1936\;\;1936 
\\1\;\;1\;\;1\;\;1\;\;1\;\;1\;\;1\;\;45\;\;45\;\;210\;\;210\;\;210\;\;210\;\;210\;\;210\;\;210\;\;1013\;\;1013 
\\10\;\;10\;\;10\;\;120\;\;120\;\;120\;\;120\;\;120\;\;120\;\;126\;\;126\;\;126^*\;\;126^*\;\;131\;\;131\;\;143\;\;714\;\;714\;\;714^*\;\;714^*
\\1\;\;1\;\;1\;\;1\;\;1\;\;10\;\;45\;\;45\;\;45\;\;131\;\;210\;\;210\;\;210\;\;210\;\;1013\;\;1013\;\;1013 
\\10\;\;10\;\;120\;\;120\;\;120\;\;120\;\;126\;\;126\;\;126^*\;\;126^*\;\;131\;\;131\;\;143\;\;1936 
\\1\;\;45\;\;210\;\;210\;\;1013 
\\120\end{array}\]

\[\begin{array}{c}
126
\\45\;\;210\;\;1013
\\10\;\;120\;\;120\;\;714
\\1\;\;1\;\;10\;\;131
\\126\;\;126^*\;\;126^*\;\;143\;\;143\;\;1936
\\1\;\;1\;\;1\;\;45\;\;210\;\;210\;\;210\;\;1013
\\10\;\;120\;\;120\;\;126^*\;\;714\;\;714^*\;\;714^*
\\1\;\;1\;\;10\;\;45\;\;131\;\;210\;\;1013
\\10\;\;10\;\;120\;\;120\;\;126\;\;126\;\;126\;\;126\;\;126^*\;\;131\;\;143\;\;143\;\;1936
\\1\;\;1\;\;1\;\;45\;\;45\;\;210\;\;210\;\;210\;\;210\;\;1013\;\;1013
\\10\;\;10\;\;120\;\;120\;\;126\;\;126^*\;\;126^*\;\;126^*\;\;131\;\;714
\\45\;\;210\;\;1013
\\126\end{array}\]

\[\begin{array}{c}
131
\\143
\\1\;\;1\;\;210\;\;1013
\\10\;\;120\;\;120\;\;120\;\;714\;\;714^*
\\1\;\;1\;\;1\;\;1\;\;10\;\;131\;\;131
\\126\;\;126^*\;\;143\;\;143\;\;143\;\;143\;\;1936
\\1\;\;1\;\;1\;\;1\;\;1\;\;45\;\;45\;\;210\;\;210\;\;210\;\;210\;\;1013
\\10\;\;10\;\;120\;\;120\;\;120\;\;120\;\;714\;\;714\;\;714^*\;\;714^*
\\1\;\;1\;\;1\;\;1\;\;10\;\;131\;\;131\;\;131\;\;131
\\1\;\;126\;\;126^*\;\;143\;\;143\;\;143\;\;210\;\;1013\;\;1936
\\1\;\;1\;\;1\;\;10\;\;10\;\;120\;\;120\;\;126\;\;126^*\;\;131\;\;131\;\;143\;\;210\;\;210\;\;1013
\\1\;\;1\;\;1\;\;45\;\;45\;\;120\;\;210\;\;210\;\;210\;\;714\;\;714^*\;\;1013
\\1\;\;10\;\;10\;\;120\;\;120\;\;126\;\;126^*\;\;131\;\;131\;\;131
\\1\;\;143\;\;210\;\;1013
\\131\end{array}\]

\[\begin{array}{c}
143
\\1
\\120\;\;714\;\;714^*
\\1\;\;1\;\;1\;\;10\;\;131
\\143\;\;143\;\;143\;\;1936\;\;1936
\\1\;\;1\;\;1\;\;1\;\;1\;\;45\;\;210\;\;210\;\;210\;\;1013
\\10\;\;10\;\;10\;\;120\;\;120\;\;120\;\;120\;\;120\;\;143\;\;714\;\;714\;\;714\;\;714\;\;714\;\;714
\\1\;\;1\;\;1\;\;1\;\;1\;\;1\;\;1\;\;1\;\;10\;\;10\;\;131\;\;131\;\;131\;\;131
\\10\;\;120\;\;126\;\;126\;\;126^*\;\;126^*\;\;143\;\;143\;\;143\;\;143\;\;143\;\;143\;\;1936\;\;1936
\\1\;\;1\;\;1\;\;1\;\;1\;\;1\;\;1\;\;1\;\;45\;\;45\;\;45\;\;210\;\;210\;\;210\;\;210\;\;210\;\;210\;\;210\;\;1013,
1013
\\10\;\;10\;\;10\;\;120\;\;120\;\;120\;\;120\;\;120\;\;143\;\;714\;\;714\;\;714\;\;714^*\;\;714^*\;\;714^*
\\1\;\;1\;\;1\;\;1\;\;1\;\;1\;\;10\;\;10\;\;131\;\;131\;\;131\;\;1013
\\10\;\;120\;\;126\;\;126\;\;126^*\;\;126^*\;\;131\;\;143\;\;143\;\;143\;\;1936\;\;1936
\\1\;\;1\;\;1\;\;1\;\;45\;\;210\;\;210\;\;210\;\;1013
\\10\;\;120\;\;143\;\;714\;\;714^*
\\1\;\;131
\\143\end{array}\]



\[\begin{array}{c}
210
\\10\;\;120\;\;120\;\;714\;\;714^*
\\1\;\;1\;\;1\;\;131
\\126\;\;126^*\;\;143\;\;143\;\;143\;\;1936\;\;1936
\\1\;\;1\;\;1\;\;1\;\;1\;\;1\;\;45\;\;45\;\;210\;\;210\;\;210\;\;210\;\;210\;\;1013\;\;1013
\\10\;\;10\;\;10\;\;120\;\;120\;\;120\;\;120\;\;120\;\;120\;\;120\;\;714\;\;714\;\;714\;\;714\;\;714^*\;\;714^*\;\;714^*\;\;714^*
\\1\;\;1\;\;1\;\;1\;\;1\;\;1\;\;1\;\;1\;\;10\;\;10\;\;10\;\;131\;\;131\;\;131\;\;131
\\126\;\;126\;\;126\;\;126^*\;\;126^*\;\;126^*\;\;143\;\;143\;\;143\;\;143\;\;143\;\;143\;\;143\;\;1936\;\;1936,
1936
\\1\;\;1\;\;1\;\;1\;\;1\;\;1\;\;1\;\;1\;\;1\;\;45\;\;45\;\;210\;\;210\;\;210\;\;210\;\;210\;\;210\;\;210\;\;210\;\;210\;\;1013\;\;1013
\\10\;\;10\;\;10\;\;120\;\;120\;\;120\;\;120\;\;120\;\;120\;\;120\;\;126\;\;126\;\;131\;\;714\;\;714\;\;714\;\;714\;\;714^*\;\;714^*\;\;714^*\;\;714^*
\\1\;\;1\;\;1\;\;1\;\;1\;\;1\;\;1\;\;10\;\;10\;\;45\;\;45\;\;131\;\;131\;\;210\;\;210\;\;210\;\;210\;\;1013\;\;1013
\\10\;\;10\;\;10\;\;120\;\;120\;\;120\;\;120\;\;126\;\;126\;\;126\;\;126\;\;126^*\;\;126^*\;\;126^*\;\;126^*\;\;131\;\;131\;\;131\;\;143\;\;143\;\;143\;\;1936\;\;1936
\\1\;\;1\;\;1\;\;1\;\;45\;\;45\;\;210\;\;210\;\;210\;\;210\;\;210\;\;210\;\;1013\;\;1013
\\10\;\;120\;\;120\;\;126\;\;126^*\;\;131\;\;714\;\; 714^*
\\210\end{array}\]





\[\begin{array}{c}
714
\\1
\\143\;\;1936
\\1\;\;1\;\;210
\\10\;\;120\;\;120\;\;714\;\;714\;\;714^*\;\;714^*
\\1\;\;1\;\;1\;\;1\;\;10\;\;131
\\126\;\;143\;\;143\;\;143\;\;1936\;\;1936
\\1\;\;1\;\;1\;\;1\;\;1\;\;45\;\;210\;\;210\;\;210\;\;210\;\;1013
\\10\;\;10\;\;120\;\;120\;\;120\;\;120\;\;714\;\;714\;\;714\;\;714^*\;\;714^*\;\;714^*
\\1\;\;1\;\;1\;\;1\;\;1\;\;10\;\;10\;\;131\;\;131
\\126\;\;126^*\;\;126^*\;\;143\;\;143\;\;143\;\;1936\;\;1936
\\1\;\;1\;\;1\;\;1\;\;45\;\;210\;\;210\;\;210\;\;210\;\;1013
\\10\;\;120\;\;120\;\;714\;\;714\;\;714^*\;\;714^*
\\1\;\;1\;\;10\;\;131
\\126\;\;143\;\;1936
\\1\;\;210
\\714\end{array}\]

\[\begin{array}{c}
1013
\\120
\\1\;\;10\;\;131
\\126\;\;126^*\;\;143\;\;1936
\\1\;\;1\;\;45\;\;45\;\;210\;\;210\;\;1013
\\10\;\;10\;\;120\;\;120\;\;120\;\;714\;\;714^*
\\1\;\;1\;\;1\;\;10\;\;131
\\126\;\;126^*\;\;143\;\;143\;\;1936
\\1\;\;1\;\;210\;\;210\;\;1013\;\;1013
\\10\;\;120\;\;120\;\;126\;\;126^*\;\;131\;\;143\;\;714\;\;714^*
\\1\;\;1\;\;1\;\;10\;\;45\;\;45\;\;131\;\;210\;\;210\;\;1013\;\;1013
\\10\;\;10\;\;10\;\;120\;\;120\;\;120\;\;126\;\;126\;\;126^*\;\;126^*\;\;131\;\;143\;\;1936
\\1\;\;1\;\;45\;\;45\;\;210\;\;210\;\;1013\;\;1013
\\10\;\;120\;\;126\;\;126^*\;\;131
\\1013\end{array}\]

\[\begin{array}{c}
1936
\\1
\\10\;\;120\;\;714\;\;714^*
\\1\;\;1
\\143\;\;143\;\;1936
\\1\;\;1\;\;1\;\;210\;\;210\;\;1013
\\10\;\;120\;\;120\;\;120\;\;714\;\;714\;\;714^*\;\;714^*
\\1\;\;1\;\;1\;\;1\;\;10\;\;10\;\;131
\\126\;\;126^*\;\;143\;\;143\;\;1936\;\;1936\;\;1936
\\1\;\;1\;\;1\;\;1\;\;45\;\;45\;\;210\;\;210\;\;210\;\;1013
\\10\;\;10\;\;120\;\;120\;\;120\;\;714\;\;714\;\;714^*\;\;714^*
\\1\;\;1\;\;1\;\;10\;\;131
\\126\;\;126^*\;\;143\;\;143\;\;1936
\\1\;\;1\;\;210\;\;210\;\;1013
\\120\;\;714\;\;714^*
\\1\;\;10
\\1936\end{array}\]

\[\begin{array}{c}
54
\\1431\;\;1728
\\54\;\;54\;\;297\;\;945
\\1431\;\;1728
\\54
\end{array}\quad
\begin{array}{c}
297
\\1728
\\54\;\;297\;\;945
\\1728
\\297
\end{array}\quad
\begin{array}{c}
945
\\1431\;\;1728
\\54\;\;297\;\;945\;\;945
\\1431\;\;1728
\\945
\end{array}\quad
\begin{array}{c}
1431
\\54\;\;945
\\1431\;\;1728
\\54\;\;945
\\1431
\end{array}\quad
\begin{array}{c}
1728
\\54\;\;945\;\;297
\\1431\;\;1728\;\;1728
\\54\;\;297\;\;945
\\1728
\end{array}
\]


Cartan matrix for block 2:

\begin{center}\begin{tabular}{c|ccccc}
&$54$&$297$&$945$&$1431$&$1728$
\\\hline $54$&4&1&1&2&2
\\$297$&1&3&1&0&2
\\$945$&1&1&4&2&2
\\$1431$&2&0&2&3&1
\\$1728$&2&2&2&1&4
\end{tabular}\end{center}

Dimensions of projectives for $\Alt(11)$:

\begin{tabular}{cccccccccc}
$1$& $10$& $34$& $109$& $120$& $131$& $210$& $320$& $714$& $791$
\\14499& 5994& 8019& 5346& 6885& 3564& 7371& 6237& 8019& 6237
\end{tabular}

\begin{tabular}{cccc}
45&126&$126^*$&693
\\2025& 2106& 2106& 2673
\end{tabular}

Block 3: 1782


Dimensions of projectives for principal block of $\Alt(12)$:

\begin{tabular}{ccccccccccccc}
$1$&$10$&$45$&$120$&$126$&$126^*$&$131$&$143$&$210$&$714$&$714^*$&$1013$&$1936$
\\110322& 48357& 18954& 51030& 19926& 19926& 24057& 37665& 50544& 30618& 30618& 24057& 29403
\end{tabular}

[ 110322, 48357, 18954, 7776, 51030, 19926, 19926, 24057, 37665, 50544, 5346,
30618, 30618, 5346, 10449, 24057, 8019, 10935, 29403, 2673, 8019 ]

Time to compute the projective using condensation subgroup of order $100$.

\begin{tabular}{ccccccc}
1 & 10 & 120 & 131 & 143 & 210 & 714
\\ 85795 & 3136 & 3601 & 1967 & 5991 & 6342 & 16409
\end{tabular}


This means that we only get 2673 when inducing from block 2.

\subsection*{Some easy projectives}

$P(1013)\oplus 2673=2673\otimes 10$

$P(45)\uparrow^G=P(45)\oplus2673^{\oplus 2}$

$P(126)\uparrow^G=P(126)\oplus2673^{\oplus 2}$

\end{document}



\newpage
\section{$F_4(2)$}

Let $246_1$ be the module inside $\Lambda^2(26_1)$.

$\Ext^1(26_1,M)=0$ for $M=1,26_1,676,4096_1,4096_2$
$\Ext^1(26_1,26_2)=2$
$\Ext^1(26_1,246_1)=2$



$676=26_1\otimes 26_2$.

So far have: 1,26,26,246,246,676,4096,4096,60516,106496,106496,

0000=1

0001=26
1000=26
0010=246
0100=246

1001=676
0011=4096
1100=4096
0101=6396
1010=6396
0110=

1110=
0111=
1101=
1011=

1111=Steinberg

\section{$F_4(3)$}

\subsection*{Block information}

\begin{center}\begin{tabular}{ccc}
\hline Block & Kernel & Modules
\\\hline $1$ & Yes & $1$, $25$, $52$, $196$, $324$, $1053_1$, $1053_2$, $1222$, $2404$, $5369$, $6707$, $7371$, $11907$
\\\hline\end{tabular}\end{center}

\subsection*{Highest weight labels}

\begin{center}\begin{tabular}{c|ccccccccccccc}
Block 1 & $1$&$25$&$52$&$196$&$324$&$1053_1$&$1053_2$&$1222$&$2404$&$5369$&$6707$&$7371$&$11907$
\\\hline  $L(\lambda)$&0000&0001&1000&0010&0002&1001&2000&0100&0011&1100&1002&1010&0101
\end{tabular}\end{center}

\subsection*{Extension information}
\begin{center}\begin{tabular}{ccc}\hline
Module & Tested to & $\Ext^1=k$
\\\hline $1$ & $2404$ & $25,2404$
\\ $25$ & $196$ & $1,196$
\\ $52$ & $52$ & None
\\ \hline\end{tabular}\end{center}


\section{$F_4(q)$}


For $p=2$, $26$ has no $1$-cohomology. $246$ has $1$-dimensional $1$-cohomology.

No self extensions of a $26$ either.

$26\otimes \bar 26=676$ is simple. Cohomology has dimension $2$.






The modules of dimensions at most $248$ for $F_4$ are $1$ (0000), $25$ (0001, $p=3$), $26$ (0001, $p\neq 3$), $26$ (1000, $p=2$), $52$ (1000, $p\neq 2$), $196$ (0010, $p=3$), $246$ (0010, $p\neq 3$) and $246$ (0100, $p=2$)


$\Lambda^2(0001)=273\oplus 1000$, $S^2(0001)=0001\oplus 0000\oplus 324$

$p=2$,
$p=3$,
$p>3$: $246,1,1$ or combination of $52$, $26$ and $1$.

$F_4$ has classes of rational $3$s, with 

Traces:

\begin{center}\begin{tabular}{c|cccccccc}
&$1$&$26$&$52$&
\\2A&$1$&$2$&$-4$
\\2B&$1$&$-6$&$20$
\\3A&$1$&$8$&$7$
\\3B&$1$&$-1$&$7$
\\3C&$1$&$-1$&$-2$
\\4A[A]
\\4B[A]
\\4C[A]
\\4D[B]
\\4E[B]
\end{tabular}\end{center}


Involutions in $E_8$ have trace $24$ and $-8$. This is kernels of size $136$ and $120$ respectively.

$1$-eigenspaces of $1$, $26$ and $52$ are $1$/$1$, $14$/$10$ and $24$/$36$.

Number of 52: 1,2,3,4 -> 52, 104, 156, 208. First two not allowed.

$156=a.26+b.52$ then $92$ trivials, so need $44$ or $28$ dimension.

$26a+52b$ becomes $14a+24b=44$ or $28$, not possible with $a+2b=6$, so not 156 in non-trivials.

Leaves $40$ trivials, $208$ from non-trivials:
$a+2b=8$,
$7a+12b=48$ or $40$, 48 leads to $b=4,a=0$, $40$ leads to contradiction
$5a+18b=48$ or $40$ $b=4,a=0$ leads to contradiction.

So number of $26$-dimensional summands is odd!!!

Subtract $26$ from everything and do again.




14 trivials, 7 26s and 1 52 seems to be only solution. This comes from $F_4G_2$, wonder if that's known.



\end{document}

\newpage
\section{Group}

\subsection*{Block information}

\begin{center}\begin{tabular}{ccc}
\hline Block & Kernel & Modules
\\\hline\end{tabular}\end{center}

\subsection*{Highest weight labels}

\begin{center}\begin{tabular}{cccccccccc}
\hline $\lambda$&
\\\hline $L(\lambda)$ &
\\ $\dim(W(\lambda))$&
\\\hline $\lambda$&
\\\hline $L(\lambda)$&
\\ $\dim(W(\lambda))$&
\\\hline
\end{tabular}\end{center}

\subsection*{Extension information}

\subsection*{Symmetric and exterior squares of modules}

\subsection*{Actions of unipotent elements on modules}

\subsection*{Traces of semisimple elements on modules}
