# M(32,51,11)

M(32,51,11) - $k(((C_2)^4 : C_{15}) \times C_2)$
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Representative: $k(((C_2)^4 : C_{15}) \times C_2)$ $(C_2)^5$ $C_{15}$, ? 32 15 1 See below. Yes Yes Yes $\mathcal{O} (((C_2)^4 : C_{15}) \times C_2)$ See below. 1 No Yes M(32,51,12) Yes

A block with defect group $(C_2)^5$ and inertial quotient $C_{15}$ is in this Morita equivalence class or in M(32,51,12), which is derived equivalent to this class.

It is unknown whether this Morita equivalence class contains blocks with inertial quotient $C_7:C_3 \times C_3$ (with action as in M(32,51,24)).

## 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(32,51,11), then $B$ is in M(32,51,2), M(32,51,5), or M(32,51,11).

## Projective indecomposable modules

Labelling the simple $B$-modules by $S_1, \dots, S_{15}$, the projective indecomposable modules have Loewy structure as follows:

$\begin{array}{ccccc} \begin{array}{c} S_1 \\ S_1 S_{10} S_{11} S_9 S_4 \\ S_{11} S_9 S_4 S_{10} S_{12} S_3 S_{14} S_{15} S_2 S_{13} \\ S_3 S_{12} S_{14} S_2 S_{15} S_{13} S_5 S_6 S_8 S_7 \\ S_6 S_5 S_7 S_8 S_1 \\ S_1 \\ \end{array} & \begin{array}{c} S_2 \\ S_{12} S_{14} S_5 S_7 S_2 \\ S_{10} S_7 S_{14} S_6 S_8 S_4 S_1 S_5 S_{12} S_{15} \\ S_{15} S_6 S_1 S_8 S_4 S_{10} S_3 S_9 S_{13} S_{11} \\ S_9 S_{13} S_3 S_{11} S_2 \\ S_2 \\ \end{array} & \begin{array}{c} S_3 \\ S_3 S_7 S_2 S_6 S_{11} \\ S_{12} S_7 S_{11} S_5 S_2 S_4 S_6 S_1 S_{14} S_{13} \\ S_{14} S_4 S_1 S_5 S_{12} S_{13} S_{15} S_9 S_8 S_{10} \\ S_9 S_8 S_{10} S_{15} S_3 \\ S_3 \\ \end{array} & \begin{array}{c} S_4 \\ S_{13} S_9 S_{14} S_{15} S_4 \\ S_{14} S_2 S_8 S_6 S_3 S_{10} S_{15} S_5 S_9 S_{13} \\ S_8 S_{10} S_1 S_5 S_2 S_7 S_3 S_6 S_{12} S_{11} \\ S_7 S_1 S_{12} S_{11} S_4 \\ S_4 \\ \end{array} & \begin{array}{c} S_5 \\ S_1 S_{10} S_{12} S_5 S_8 \\ S_4 S_7 S_1 S_{10} S_8 S_9 S_3 S_{15} S_{12} S_{11} \\ S_{14} S_{11} S_{13} S_2 S_4 S_7 S_{15} S_3 S_9 S_6 \\ S_{13} S_{14} S_6 S_5 S_2 \\ S_5 \\ \end{array} \end{array}$

$\begin{array}{ccccc} \begin{array}{c} S_6 \\ S_1 S_5 S_{11} S_6 S_{13} \\ S_{12} S_{11} S_{10} S_1 S_5 S_2 S_8 S_9 S_4 S_{13} \\ S_9 S_3 S_{10} S_{14} S_{15} S_8 S_7 S_2 S_4 S_{12} \\ S_7 S_{14} S_{15} S_3 S_6 \\ S_6 \\ \end{array} & \begin{array}{c} S_7 \\ S_4 S_6 S_{14} S_1 S_7 \\ S_5 S_9 S_{13} S_{14} S_4 S_6 S_{11} S_{15} S_1 S_{10} \\ S_3 S_{13} S_8 S_2 S_9 S_{12} S_{15} S_5 S_{11} S_{10} \\ S_{12} S_2 S_7 S_3 S_8 \\ S_7 \\ \end{array} & \begin{array}{c} S_8 \\ S_8 S_1 S_3 S_7 S_9 \\ S_9 S_{14} S_1 S_4 S_{10} S_6 S_2 S_{11} S_7 S_3 \\ S_{14} S_{15} S_6 S_2 S_4 S_{11} S_{10} S_{13} S_{12} S_5 \\ S_{15} S_5 S_{13} S_8 S_{12} \\ S_8 \\ \end{array} & \begin{array}{c} S_9 \\ S_3 S_{14} S_{10} S_2 S_9 \\ S_{10} S_{15} S_2 S_7 S_6 S_{14} S_{12} S_{11} S_5 S_3 \\ S_4 S_6 S_5 S_{12} S_{13} S_{11} S_7 S_1 S_{15} S_8 \\ S_1 S_{13} S_8 S_4 S_9 \\ S_9 \\ \end{array} & \begin{array}{c} S_{10} \\ S_{15} S_{12} S_3 S_{10} S_{11} \\ S_4 S_2 S_{13} S_8 S_7 S_6 S_3 S_{15} S_{11} S_{12} \\ S_9 S_5 S_7 S_{14} S_6 S_{13} S_2 S_8 S_4 S_1 \\ S_{14} S_5 S_9 S_1 S_{10} \\ S_{10} \\ \end{array} \end{array}$

$\begin{array}{ccccc} \begin{array}{c} S_{11} \\ S_{13} S_{11} S_2 S_{12} S_4 \\ S_8 S_4 S_9 S_7 S_{12} S_{15} S_2 S_{13} S_{14} S_5 \\ S_1 S_{14} S_8 S_3 S_5 S_9 S_6 S_{15} S_{10} S_7 \\ S_1 S_{10} S_6 S_3 S_{11} \\ S_{11} \\ \end{array} & \begin{array}{c} S_{12} \\ S_{15} S_4 S_{12} S_7 S_8 \\ S_6 S_3 S_4 S_9 S_1 S_7 S_{14} S_{13} S_{15} S_8 \\ S_2 S_{13} S_3 S_1 S_9 S_6 S_{10} S_{11} S_{14} S_5 \\ S_{11} S_5 S_2 S_{10} S_{12} \\ S_{12} \\ \end{array} & \begin{array}{c} S_{13} \\ S_8 S_5 S_{13} S_9 S_2 \\ S_3 S_{12} S_7 S_1 S_5 S_{10} S_9 S_8 S_2 S_{14} \\ S_{12} S_7 S_3 S_{10} S_{14} S_1 S_{11} S_{15} S_6 S_4 \\ S_{11} S_4 S_{15} S_6 S_{13} \\ S_{13} \\ \end{array} & \begin{array}{c} S_{14} \\ S_6 S_{10} S_{14} S_{15} S_5 \\ S_{15} S_5 S_3 S_{12} S_{13} S_{11} S_1 S_{10} S_8 S_6 \\ S_2 S_{12} S_8 S_{13} S_3 S_1 S_{11} S_4 S_9 S_7 \\ S_7 S_4 S_9 S_2 S_{14} \\ S_{14} \\ \end{array} & \begin{array}{c} S_{15} \\ S_{15} S_6 S_8 S_{13} S_3 \\ S_2 S_8 S_{13} S_3 S_5 S_1 S_6 S_9 S_{11} S_7 \\ S_2 S_{11} S_7 S_5 S_1 S_9 S_{10} S_4 S_{12} S_{14} \\ S_{14} S_{12} S_{10} S_4 S_{15} \\ S_{15} \\ \end{array} \end{array}$

## Irreducible characters

All irreducible characters have height zero.

## Cartan matrix

$\left( \begin{array}{ccc} 4 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \\ 2 & 4 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \\ 2 & 2 & 4 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \\ 2 & 2 & 2 & 4 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \\ 2 & 2 & 2 & 2 & 4 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \\ 2 & 2 & 2 & 2 & 2 & 4 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \\ 2 & 2 & 2 & 2 & 2 & 2 & 4 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \\ 2 & 2 & 2 & 2 & 2 & 2 & 2 & 4 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \\ 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 4 & 2 & 2 & 2 & 2 & 2 & 2 \\ 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 4 & 2 & 2 & 2 & 2 & 2 \\ 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 4 & 2 & 2 & 2 & 2 \\ 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 4 & 2 & 2 & 2 \\ 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 4 & 2 & 2 \\ 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 4 & 2 \\ 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 4 \end{array} \right)$

## Decomposition matrix

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