# M(32,51,21)

M(32,51,21) - $B_0(k((SL_2(8) \times (C_2)^2) : C_3))$
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Representative: $B_0(k((SL_2(8) \times (C_2)^2) : C_3))$ $(C_2)^5$ $C_7:C_3$ 16 5 1 $\left( \begin{array}{ccccc} 12 & 10 & 10 & 16 & 8 \\ 10 & 12 & 10 & 16 & 8 \\ 10 & 10 & 12 & 16 & 8 \\ 16 & 16 & 16 & 32 & 12 \\ 8 & 8 & 8 & 12 & 8 \end{array} \right)$ Yes Yes Yes $B_0(\mathcal{O}((SL_2(8) \times (C_2)^2) : C_3))$ See below. 1 No Yes M(32,51,20) 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(32,51,21), then $B$ is in M(32,51,7), M(32,51,21), or M(32,51,28).

## Projective indecomposable modules

Labelling the simple $B$-modules by $S_1, \dots, S_5$, the projective indecomposable modules have Loewy structure as follows:

$\begin{array}{ccc} \begin{array}{c} S_{1} \\ S_{3} S_{2} S_{4} \\ S_{3} S_{2} S_{1} S_{1} S_{4} S_{4} S_{5} \\ S_{2} S_{2} S_{3} S_{1} S_{3} S_{1} S_{4} S_{4} S_{4} S_{5} S_{5} \\ S_{3} S_{2} S_{3} S_{1} S_{1} S_{2} S_{4} S_{4} S_{4} S_{4} S_{5} S_{5} \\ S_{3} S_{1} S_{2} S_{2} S_{1} S_{3} S_{4} S_{4} S_{4} S_{5} S_{5} \\ S_{3} S_{1} S_{2} S_{1} S_{4} S_{4} S_{5} \\ S_{2} S_{3} S_{4} \\ S_{1} \\ \end{array} & \begin{array}{c} S_{2} \\ S_{3} S_{1} S_{4} \\ S_{1} S_{2} S_{2} S_{3} S_{4} S_{4} S_{5} \\ S_{1} S_{2} S_{3} S_{3} S_{2} S_{1} S_{4} S_{4} S_{4} S_{5} S_{5} \\ S_{3} S_{1} S_{3} S_{2} S_{1} S_{2} S_{4} S_{4} S_{4} S_{4} S_{5} S_{5} \\ S_{3} S_{1} S_{1} S_{2} S_{2} S_{3} S_{4} S_{4} S_{4} S_{5} S_{5} \\ S_{3} S_{1} S_{2} S_{2} S_{4} S_{4} S_{5} \\ S_{3} S_{1} S_{4} \\ S_{2} \\ \end{array} & \begin{array}{c} S_{3} \\ S_{2} S_{1} S_{4} \\ S_{1} S_{3} S_{3} S_{2} S_{4} S_{4} S_{5} \\ S_{2} S_{3} S_{1} S_{3} S_{1} S_{2} S_{4} S_{4} S_{4} S_{5} S_{5} \\ S_{2} S_{1} S_{3} S_{3} S_{2} S_{1} S_{4} S_{4} S_{4} S_{4} S_{5} S_{5} \\ S_{3} S_{1} S_{1} S_{2} S_{2} S_{3} S_{4} S_{4} S_{4} S_{5} S_{5} \\ S_{1} S_{2} S_{3} S_{3} S_{4} S_{4} S_{5} \\ S_{1} S_{2} S_{4} \\ S_{3} \\ \end{array} \end{array}$

$\begin{array}{cc} \begin{array}{c} S_{4} \\ S_{2} S_{3} S_{1} S_{4} S_{4} S_{5} \\ S_{1} S_{2} S_{3} S_{3} S_{1} S_{2} S_{4} S_{4} S_{4} S_{4} S_{5} S_{5} \\ S_{2} S_{2} S_{1} S_{1} S_{1} S_{3} S_{3} S_{3} S_{2} S_{4} S_{4} S_{4} S_{4} S_{4} S_{4} S_{5} S_{5} \\ S_{1} S_{3} S_{2} S_{1} S_{3} S_{2} S_{2} S_{3} S_{2} S_{3} S_{1} S_{1} S_{4} S_{4} S_{4} S_{4} S_{4} S_{4} S_{5} S_{5} \\ S_{2} S_{3} S_{3} S_{1} S_{1} S_{2} S_{1} S_{2} S_{3} S_{4} S_{4} S_{4} S_{4} S_{4} S_{4} S_{5} S_{5} \\ S_{3} S_{2} S_{1} S_{1} S_{2} S_{3} S_{4} S_{4} S_{4} S_{4} S_{5} S_{5} \\ S_{3} S_{1} S_{2} S_{4} S_{4} S_{5} \\ S_{4} \\ \end{array} & \begin{array}{c} S_{5} \\ S_{4} S_{5} S_{5} \\ S_{3} S_{2} S_{1} S_{4} S_{4} S_{5} \\ S_{1} S_{1} S_{2} S_{3} S_{3} S_{2} S_{4} S_{4} \\ S_{1} S_{2} S_{1} S_{3} S_{2} S_{3} S_{4} S_{4} \\ S_{2} S_{3} S_{1} S_{3} S_{1} S_{2} S_{4} S_{4} \\ S_{3} S_{1} S_{2} S_{4} S_{4} S_{5} \\ S_{4} S_{5} S_{5} \\ S_{5} \\ \end{array} \end{array}$

## Irreducible characters

All irreducible characters have height zero.

## Decomposition matrix

$\left( \begin{array}{ccccc} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 \\ 1 & 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & 1 & 0 \\ 0 & 1 & 0 & 1 & 0 \\ 1 & 1 & 1 & 3 & 0 \\ 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 2 & 1 \\ 1 & 1 & 1 & 2 & 1 \\ 1 & 1 & 1 & 2 & 1 \\ 1 & 1 & 1 & 2 & 1 \end{array}\right)$