# M(32,51,13)

M(32,51,13) - $k(((C_2)^3 : C_7) \times A_4)$
[[File: |250px]]
Representative: $k(((C_2)^3 : C_7) \times A_4)$ $(C_2)^5$ $C_{21}$ 32 21 1 See below. Yes Yes Yes $\mathcal{O} (((C_2)^3 : C_7) \times A_4)$ See below. 1 No Yes M(32,51,14), M(32,51,15), M(32,51,16) Yes

## Covering blocks and covered blocks

Let $N \triangleleft G$ with prime $p'$-index and let $B$ be a block of $\mathcal{O} G$ covering a block $b$ of $\mathcal{O} N$.

If $b$ is in M(32,51,13), then $B$ is in M(32,51,2), M(32,51,6), M(32,51,13), M(32,51,24) or M(31,51,33).

## Projective indecomposable modules

Labelling the simple $B$-modules by $S_1, \dots, S_{21}$, the projective indecomposable modules have Loewy structure as follows:

$\begin{array}{cccc} \begin{array}{c} S_{1} \\ S_{20} S_{3} S_{19} S_{2} S_{21} \\ S_{14} S_{12} S_{4} S_{5} S_{16} S_{13} S_{18} S_{15} S_{17} S_{1} \\ S_{10} S_{3} S_{20} S_{7} S_{9} S_{19} S_{6} S_{1} S_{8} S_{11} \\ S_{2} S_{21} S_{13} S_{12} S_{5} \\ S_{1} \\ \end{array} & \begin{array}{c} S_{2} \\ S_{15} S_{4} S_{17} S_{1} S_{21} \\ S_{18} S_{8} S_{3} S_{6} S_{20} S_{14} S_{16} S_{19} S_{10} S_{2} \\ S_{7} S_{15} S_{17} S_{11} S_{2} S_{4} S_{5} S_{12} S_{13} S_{9} \\ S_{1} S_{21} S_{10} S_{8} S_{6} \\ S_{2} \\ \end{array} & \begin{array}{c} S_{3} \\ S_{13} S_{5} S_{19} S_{4} S_{16} \\ S_{3} S_{9} S_{11} S_{20} S_{17} S_{12} S_{1} S_{14} S_{8} S_{6} \\ S_{15} S_{7} S_{13} S_{5} S_{2} S_{3} S_{19} S_{10} S_{18} S_{21} \\ S_{16} S_{4} S_{1} S_{20} S_{12} \\ S_{3} \\ \end{array} & \begin{array}{c} S_{4} \\ S_{8} S_{6} S_{17} S_{16} S_{3} \\ S_{2} S_{15} S_{14} S_{10} S_{5} S_{13} S_{19} S_{9} S_{11} S_{4} \\ S_{1} S_{18} S_{20} S_{6} S_{8} S_{17} S_{4} S_{12} S_{7} S_{21} \\ S_{3} S_{16} S_{2} S_{15} S_{10} \\ S_{4} \\ \end{array} \end{array}$

$\begin{array}{cccc} \begin{array}{c} S_{5} \\ S_{1} S_{20} S_{13} S_{9} S_{6} \\ S_{5} S_{8} S_{2} S_{18} S_{12} S_{3} S_{19} S_{21} S_{11} S_{15} \\ S_{13} S_{7} S_{4} S_{20} S_{17} S_{5} S_{10} S_{14} S_{16} S_{1} \\ S_{6} S_{9} S_{12} S_{3} S_{19} \\ S_{5} \\ \end{array} & \begin{array}{c} S_{6} \\ S_{8} S_{15} S_{2} S_{5} S_{9} \\ S_{4} S_{11} S_{20} S_{21} S_{10} S_{13} S_{1} S_{17} S_{18} S_{6} \\ S_{2} S_{12} S_{7} S_{3} S_{8} S_{6} S_{15} S_{16} S_{19} S_{14} \\ S_{5} S_{9} S_{4} S_{10} S_{17} \\ S_{6} \\ \end{array} & \begin{array}{c} S_{7} \\ S_{9} S_{16} S_{21} S_{12} S_{10} \\ S_{5} S_{3} S_{11} S_{1} S_{6} S_{18} S_{4} S_{14} S_{2} S_{7} \\ S_{9} S_{7} S_{20} S_{15} S_{17} S_{16} S_{21} S_{8} S_{19} S_{13} \\ S_{10} S_{12} S_{18} S_{11} S_{14} \\ S_{7} \\ \end{array} & \begin{array}{c} S_{8} \\ S_{2} S_{17} S_{10} S_{11} S_{13} \\ S_{12} S_{7} S_{21} S_{14} S_{1} S_{15} S_{6} S_{19} S_{4} S_{8} \\ S_{16} S_{18} S_{10} S_{3} S_{17} S_{8} S_{5} S_{20} S_{9} S_{2} \\ S_{13} S_{11} S_{4} S_{15} S_{6} \\ S_{8} \\ \end{array} \end{array}$

$\begin{array}{cccc} \begin{array}{c} S_{9} \\ S_{21} S_{11} S_{18} S_{5} S_{6} \\ S_{13} S_{14} S_{15} S_{20} S_{8} S_{16} S_{7} S_{2} S_{1} S_{9} \\ S_{12} S_{3} S_{10} S_{4} S_{11} S_{9} S_{19} S_{17} S_{21} S_{18} \\ S_{6} S_{5} S_{7} S_{16} S_{14} \\ S_{9} \\ \end{array} & \begin{array}{c} S_{10} \\ S_{6} S_{2} S_{4} S_{12} S_{7} \\ S_{8} S_{5} S_{15} S_{16} S_{1} S_{9} S_{3} S_{21} S_{17} S_{10} \\ S_{10} S_{6} S_{18} S_{2} S_{14} S_{13} S_{20} S_{4} S_{11} S_{19} \\ S_{7} S_{12} S_{17} S_{15} S_{8} \\ S_{10} \\ \end{array} & \begin{array}{c} S_{11} \\ S_{7} S_{21} S_{14} S_{8} S_{13} \\ S_{17} S_{12} S_{10} S_{18} S_{2} S_{19} S_{1} S_{9} S_{16} S_{11} \\ S_{11} S_{5} S_{21} S_{7} S_{4} S_{20} S_{14} S_{15} S_{3} S_{6} \\ S_{8} S_{13} S_{18} S_{9} S_{16} \\ S_{11} \\ \end{array} & \begin{array}{c} S_{12} \\ S_{5} S_{1} S_{3} S_{10} S_{7} \\ S_{16} S_{20} S_{19} S_{21} S_{2} S_{4} S_{9} S_{13} S_{6} S_{12} \\ S_{14} S_{18} S_{11} S_{15} S_{17} S_{1} S_{8} S_{5} S_{3} S_{12} \\ S_{7} S_{10} S_{19} S_{13} S_{20} \\ S_{12} \\ \end{array} \end{array}$

$\begin{array}{cccc} \begin{array}{c} S_{13} \\ S_{1} S_{12} S_{19} S_{8} S_{11} \\ S_{2} S_{5} S_{21} S_{10} S_{20} S_{7} S_{3} S_{14} S_{17} S_{13} \\ S_{19} S_{16} S_{4} S_{12} S_{15} S_{9} S_{1} S_{18} S_{13} S_{6} \\ S_{8} S_{11} S_{3} S_{20} S_{5} \\ S_{13} \\ \end{array} & \begin{array}{c} S_{14} \\ S_{7} S_{9} S_{18} S_{17} S_{19} \\ S_{14} S_{6} S_{5} S_{21} S_{20} S_{11} S_{16} S_{12} S_{15} S_{10} \\ S_{13} S_{3} S_{1} S_{18} S_{7} S_{8} S_{4} S_{14} S_{9} S_{2} \\ S_{17} S_{19} S_{11} S_{21} S_{16} \\ S_{14} \\ \end{array} & \begin{array}{c} S_{15} \\ S_{4} S_{10} S_{8} S_{20} S_{18} \\ S_{12} S_{17} S_{13} S_{11} S_{3} S_{7} S_{6} S_{16} S_{2} S_{15} \\ S_{5} S_{4} S_{14} S_{1} S_{9} S_{19} S_{15} S_{10} S_{8} S_{21} \\ S_{20} S_{18} S_{17} S_{6} S_{2} \\ S_{15} \\ \end{array} & \begin{array}{c} S_{16} \\ S_{14} S_{11} S_{9} S_{3} S_{4} \\ S_{7} S_{13} S_{21} S_{19} S_{6} S_{17} S_{18} S_{5} S_{8} S_{16} \\ S_{15} S_{16} S_{14} S_{11} S_{9} S_{1} S_{12} S_{10} S_{20} S_{2} \\ S_{3} S_{4} S_{18} S_{21} S_{7} \\ S_{16} \\ \end{array} \end{array}$

$\begin{array}{ccccc} \begin{array}{c} S_{17} \\ S_{15} S_{10} S_{6} S_{19} S_{14} \\ S_{17} S_{18} S_{12} S_{8} S_{5} S_{4} S_{2} S_{20} S_{7} S_{9} \\ S_{3} S_{11} S_{21} S_{15} S_{10} S_{1} S_{17} S_{16} S_{6} S_{13} \\ S_{14} S_{19} S_{2} S_{8} S_{4} \\ S_{17} \\ \end{array} & \begin{array}{c} S_{18} \\ S_{16} S_{11} S_{7} S_{20} S_{15} \\ S_{13} S_{14} S_{9} S_{4} S_{8} S_{3} S_{12} S_{10} S_{21} S_{18} \\ S_{7} S_{6} S_{2} S_{18} S_{1} S_{11} S_{5} S_{17} S_{16} S_{19} \\ S_{15} S_{20} S_{9} S_{14} S_{21} \\ S_{18} \\ \end{array} & \begin{array}{c} S_{19} \\ S_{12} S_{20} S_{5} S_{14} S_{17} \\ S_{13} S_{7} S_{6} S_{9} S_{3} S_{10} S_{18} S_{1} S_{15} S_{19} \\ S_{8} S_{2} S_{21} S_{4} S_{12} S_{20} S_{19} S_{16} S_{11} S_{5} \\ S_{17} S_{14} S_{3} S_{1} S_{13} \\ S_{19} \\ \end{array} & \begin{array}{c} S_{20} \\ S_{12} S_{3} S_{13} S_{18} S_{15} \\ S_{5} S_{8} S_{10} S_{4} S_{11} S_{7} S_{16} S_{1} S_{19} S_{20} \\ S_{2} S_{6} S_{9} S_{14} S_{12} S_{3} S_{13} S_{20} S_{17} S_{21} \\ S_{15} S_{18} S_{19} S_{1} S_{5} \\ S_{20} \\ \end{array} & \begin{array}{c} S_{21} \\ S_{18} S_{16} S_{14} S_{1} S_{2} \\ S_{9} S_{7} S_{20} S_{15} S_{11} S_{17} S_{19} S_{4} S_{3} S_{21} \\ S_{6} S_{10} S_{14} S_{21} S_{12} S_{16} S_{5} S_{8} S_{18} S_{13} \\ S_{2} S_{1} S_{9} S_{7} S_{11} \\ S_{21} \\ \end{array} \end{array}$

## Irreducible characters

All irreducible characters have height zero.

## Cartan matrix

$\left( \begin{array}{ccccccccccccccccccccc} 4 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 2 & 4 & 2 & 1 & 1 & 1 & 1 & 1 & 1 & 2 & 1 & 1 & 2 & 2 & 1 & 1 & 2 & 2 & 1 & 1 & 2 \\ 2 & 2 & 4 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 2 & 2 & 1 & 1 & 2 & 2 & 1 & 1 & 2 & 2 & 1 \\ 2 & 1 & 1 & 4 & 2 & 2 & 2 & 2 & 2 & 1 & 1 & 1 & 2 & 1 & 1 & 1 & 1 & 1 & 2 & 1 & 1 \\ 2 & 1 & 1 & 2 & 4 & 2 & 2 & 2 & 2 & 2 & 1 & 1 & 1 & 1 & 2 & 1 & 1 & 1 & 1 & 1 & 1 \\ 2 & 1 & 1 & 2 & 2 & 4 & 2 & 2 & 2 & 1 & 1 & 1 & 1 & 1 & 1 & 2 & 1 & 1 & 1 & 1 & 2 \\ 2 & 1 & 1 & 2 & 2 & 2 & 4 & 2 & 2 & 1 & 2 & 1 & 1 & 1 & 1 & 1 & 2 & 1 & 1 & 1 & 1 \\ 2 & 1 & 1 & 2 & 2 & 2 & 2 & 4 & 2 & 1 & 1 & 2 & 1 & 1 & 1 & 1 & 1 & 2 & 1 & 1 & 1 \\ 2 & 1 & 1 & 2 & 2 & 2 & 2 & 2 & 4 & 1 & 1 & 1 & 1 & 2 & 1 & 1 & 1 & 1 & 1 & 2 & 1 \\ 1 & 2 & 1 & 1 & 2 & 1 & 1 & 1 & 1 & 4 & 1 & 1 & 2 & 2 & 2 & 1 & 2 & 2 & 1 & 1 & 2 \\ 1 & 1 & 2 & 1 & 1 & 1 & 2 & 1 & 1 & 1 & 4 & 2 & 1 & 1 & 2 & 2 & 2 & 1 & 2 & 2 & 1 \\ 1 & 1 & 2 & 1 & 1 & 1 & 1 & 2 & 1 & 1 & 2 & 4 & 1 & 1 & 2 & 2 & 1 & 2 & 2 & 2 & 1 \\ 1 & 2 & 1 & 2 & 1 & 1 & 1 & 1 & 1 & 2 & 1 & 1 & 4 & 2 & 1 & 1 & 2 & 2 & 2 & 1 & 2 \\ 1 & 2 & 1 & 1 & 1 & 1 & 1 & 1 & 2 & 2 & 1 & 1 & 2 & 4 & 1 & 1 & 2 & 2 & 1 & 2 & 2 \\ 1 & 1 & 2 & 1 & 2 & 1 & 1 & 1 & 1 & 2 & 2 & 2 & 1 & 1 & 4 & 2 & 1 & 1 & 2 & 2 & 1 \\ 1 & 1 & 2 & 1 & 1 & 2 & 1 & 1 & 1 & 1 & 2 & 2 & 1 & 1 & 2 & 4 & 1 & 1 & 2 & 2 & 2 \\ 1 & 2 & 1 & 1 & 1 & 1 & 2 & 1 & 1 & 2 & 2 & 1 & 2 & 2 & 1 & 1 & 4 & 2 & 1 & 1 & 2 \\ 1 & 2 & 1 & 1 & 1 & 1 & 1 & 2 & 1 & 2 & 1 & 2 & 2 & 2 & 1 & 1 & 2 & 4 & 1 & 1 & 2 \\ 1 & 1 & 2 & 2 & 1 & 1 & 1 & 1 & 1 & 1 & 2 & 2 & 2 & 1 & 2 & 2 & 1 & 1 & 4 & 2 & 1 \\ 1 & 1 & 2 & 1 & 1 & 1 & 1 & 1 & 2 & 1 & 2 & 2 & 1 & 2 & 2 & 2 & 1 & 1 & 2 & 4 & 1 \\ 1 & 2 & 1 & 1 & 1 & 2 & 1 & 1 & 1 & 2 & 1 & 1 & 2 & 2 & 1 & 2 & 2 & 2 & 1 & 1 & 4 \end{array}\right)$

## Decomposition matrix

$\left( \begin{array}{ccccccccccccccccccccc} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 \\ 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \end{array}\right)$