# M(32,51,2)

M(32,51,2) - $k(A_4 \times (C_2)^3)$
[[File: |250px]]
Representative: $k(A_4 \times (C_2)^3)$ $(C_2)^5$ $C_3$ 32 3 1 $\left( \begin{array}{ccc} 16 & 8 & 8\\ 8 & 16 & 8\\ 8 & 8 & 16 \end{array} \right)$ Yes Yes Yes $\mathcal{O} (A_4 \times (C_2)^3)$ See below. 1 $S_3 \times (C_2)^3 : GL_3(2)$ No Yes M(32,51,3) 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(32,51,2), then $B$ is in M(32,51,1), M(32,51,2), M(32,51,8), M(31,51,13) or M(32,51,32).

## Projective indecomposable modules

Labelling the simple $B$-modules by $S_1, S_2, S_3$, the projective indecomposable modules have Loewy structure as follows:

$\begin{array}{ccc} \begin{array}{c} S_1 \\ S_1 S_1 S_1 S_2 S_3 \\ S_1 S_1 S_1 S_1 S_2 S_2 S_2 S_3 S_3 S_3 \\ S_1 S_1 S_1 S_1 S_2 S_2 S_2 S_3 S_3 S_3 \\ S_1 S_1 S_1 S_2 S_3 \\ S_1 \\ \end{array} & \begin{array}{c} S_2 \\ S_1 S_2 S_2 S_2 S_3 \\ S_1 S_1 S_1 S_2 S_2 S_2 S_2 S_3 S_3 S_3 \\ S_1 S_1 S_1 S_2 S_2 S_2 S_2 S_3 S_3 S_3 \\ S_1 S_2 S_2 S_2 S_3 \\ S_2 \\ \end{array} & \begin{array}{c} S_3 \\ S_1 S_2 S_3 S_3 S_3 \\ S_1 S_1 S_1 S_2 S_2 S_2 S_3 S_3 S_3 S_3 \\ S_1 S_1 S_1 S_2 S_2 S_2 S_3 S_3 S_3 S_3 \\ S_1 S_2 S_3 S_3 S_3 \\ S_3 \\ \end{array} \end{array}$

## Irreducible characters

All irreducible characters have height zero.

## Decomposition matrix

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