# M(32,51,4)

M(32,51,4) - $k((C_2)^4 : C_3) \times C_2)$
[[File: |250px]]
Representative: $k((C_2)^4 : C_3) \times C_2)$ $(C_2)^5$ $C_3$ 16 3 1 $\left( \begin{array}{ccc} 12 & 10 & 10\\ 10 & 12 & 10\\ 10 & 10 & 12 \end{array} \right)$ Yes Yes Yes $\mathcal{O} ((C_2)^4 : C_3) \times C_2)$ See below. 1 No Yes Forms a derived equivalence class 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(32,51,4), then $B$ is in M(32,51,1), M(32,51,4), or M(32,51,8).

## Projective indecomposable modules

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

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

## Irreducible characters

All irreducible characters have height zero.

## Decomposition matrix

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