# M(32,51,1)

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M(32,51,1) - $k((C_2)^5)$
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Representative: $k((C_2)^5)$ $(C_2)^5$ $1$ 32 1 1 $\left( \begin{array}{c} 32 \end{array} \right)$ Yes Yes Yes $\mathcal{O} ((C_2)^5)$ $\left( \begin{array}{c} 1 \\ 1 \\ \vdots \\ 1 \end{array}\right)$ 1 $(C_2)^5:GL_5(2)$ No Yes Forms a derived equivalence class Yes

These are nilpotent blocks.

## Covering blocks and covered blocks

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

If $b$ is in M(32,51,1), then $B$ is in M(32,51,1), M(32,51,2), M(32,51,4), M(32,51,6), M(32,51,8), M(16,14,11), M(16,14,13), M(16,14,17), M(16,14,20),M(16,14,22),M(16,14,24) or M(16,14,30).

## Projective indecomposable modules

Labelling the unique simple $B$-module by $S_1$, the unique projective indecomposable module has Loewy structure as follows:

$\begin{array}{c} S_1 \\ S_1 S_1 S_1 S_1 S_1 \\ S_1 S_1 S_1 S_1 S_1 S_1 S_1 S_1 S_1 S_1 \\ S_1 S_1 S_1 S_1 S_1 S_1 S_1 S_1 S_1 S_1 \\ S_1 S_1 S_1 S_1 S_1 \\ S_1 \\ \end{array}$

## Irreducible characters

All irreducible characters have height zero.