M(32,51,5)

M(32,51,5) - $k((C_2)^4 : C_5) \times C_2)$
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Representative: $k((C_2)^4 : C_5) \times C_2)$ $(C_2)^5$ $C_5$, ? 16 5 1 $\left( \begin{array}{ccc} 8 & 6 & 6 & 6 & 6 \\ 6 & 8 & 6 & 6 & 6 \\ 6 & 6 & 8 & 6 & 6 \\ 6 & 6 & 6 & 8 & 6 \\ 6 & 6 & 6 & 6 & 8 \end{array} \right)$ Yes Yes Yes $\mathcal{O} ((C_2)^4 : C_5) \times C_2)$ See below. 1 No Yes Forms a derived equivalence class Yes

A block with defect group $(C_2)^5$ and inertial quotient $C_5$ is in this Morita equivalence class.

It is unknown whether this Morita equivalence class contains blocks with inertial quotient $C_7:C_3$ (with action as in M(32,51,20)).

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(32,51,5), then $B$ is in M(32,51,1), M(32,51,5), or M(32,51,11).

Projective indecomposable modules

Labelling the simple $B$-modules by $S_1, S_2, S_3, S_4, S_5$, the projective indecomposable modules have Loewy structure as follows:

$\begin{array}{ccccc} \begin{array}{c} S_1 \\ S_1 S_5 S_2 S_4 S_3 \\ S_3 S_5 S_4 S_2 S_1 S_4 S_3 S_1 S_5 S_2 \\ S_2 S_2 S_3 S_5 S_4 S_1 S_3 S_5 S_1 S_4 \\ S_4 S_5 S_2 S_3 S_1 \\ S_1 \\ \end{array} & \begin{array}{c} S_2 \\ S_4 S_2 S_1 S_5 S_3 \\ S_5 S_1 S_3 S_1 S_4 S_4 S_2 S_2 S_5 S_3 \\ S_2 S_3 S_1 S_5 S_4 S_2 S_3 S_4 S_5 S_1 \\ S_3 S_4 S_1 S_5 S_2 \\ S_2 \\ \end{array} & \begin{array}{c} S_3 \\ S_5 S_4 S_2 S_3 S_1 \\ S_4 S_2 S_3 S_3 S_4 S_1 S_2 S_1 S_5 S_5 \\ S_4 S_2 S_3 S_2 S_1 S_1 S_4 S_5 S_5 S_3 \\ S_5 S_2 S_4 S_1 S_3 \\ S_3 \\ \end{array} & \begin{array}{c} S_4 \\ S_5 S_1 S_2 S_3 S_4 \\ S_1 S_3 S_2 S_2 S_3 S_5 S_5 S_1 S_4 S_4 \\ S_5 S_3 S_4 S_4 S_3 S_2 S_5 S_2 S_1 S_1 \\ S_3 S_5 S_2 S_1 S_4 \\ S_4 \\ \end{array} & \begin{array}{c} S_5 \\ S_2 S_4 S_5 S_1 S_3 \\ S_5 S_3 S_5 S_2 S_4 S_2 S_3 S_4 S_1 S_1 \\ S_2 S_4 S_1 S_5 S_2 S_5 S_1 S_4 S_3 S_3 \\ S_2 S_3 S_1 S_4 S_5 \\ S_5 \\ \end{array} \end{array}$

Irreducible characters

All irreducible characters have height zero.

Decomposition matrix

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