# M(32,51,32)

M(32,51,32) - $b_2(k((C_2)^4:3^{1+2}_+) \times C_2)$
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Representative: $b_2(k((C_2)^4:3^{1+2}_+) \times C_2)$ $(C_2)^5$ $C_3 \times C_3$ 16 1 1 $\left( \begin{array}{c} 32 \end{array} \right)$ Yes Yes Yes $b_2(\mathcal{O}((C_2)^4:3^{1+2}_+) \times C_2)$ $\left( \begin{array}{c} 1 \\ 1 \\ \vdots \\ 1 \\ 3 \\ 3 \end{array}\right)$ 1 No Yes Forms a derived equivalence class Yes

This Morita equivalence class contains only non-principal blocks.

## Other notatable representatives

Any nonprincipal block of $(C_2)^4:3^{1+2}_- \times C_2$.

## 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(32,51,32), then $B$ is in M(32,51,2) or M(32,51,32).

## Projective indecomposable modules

Labelling the unique simple $b_2$-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.