# M(32,51,25)

M(32,51,25) - $B_0(k(((C_2)^3 : (C_7:C_3)) \times A_5))$
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Representative: $B_0(k(((C_2)^3 : (C_7:C_3)) \times A_5))$ $(C_2)^5$ $(C_{7}:C_3) \times C_3$ 32 15 1 See below. Yes Yes Yes $B_0(\mathcal{O}(((C_2)^3 : (C_7:C_3)) \times A_5))$ See below. 1 No Yes M(32,51,24), M(32,51,26), M(32,51,27), M(32,51,28), M(32,51,29) 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(32,51,25), then $B$ is in M(32,51,14) or M(32,51,25).

## Projective indecomposable modules

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

$\begin{array}{ccc} \begin{array}{c} S_{1} \\ S_{6} S_{7} S_{10} \\ S_{1} S_{1} S_{11} S_{14} S_{15} \\ S_{1} S_{7} S_{6} S_{10} S_{10} S_{12} S_{13} \\ S_{1} S_{7} S_{6} S_{11} S_{11} S_{14} S_{15} \\ S_{1} S_{1} S_{10} S_{12} S_{13} \\ S_{7} S_{6} S_{11} \\ S_{1} \\ \end{array} & \begin{array}{c} S_{2} \\ S_{9} S_{5} S_{10} \\ S_{2} S_{2} S_{11} S_{15} S_{14} \\ S_{2} S_{5} S_{9} S_{10} S_{10} S_{13} S_{12} \\ S_{2} S_{5} S_{9} S_{11} S_{11} S_{15} S_{14} \\ S_{2} S_{2} S_{10} S_{13} S_{12} \\ S_{9} S_{5} S_{11} \\ S_{2} \\ \end{array} & \begin{array}{c} S_{3} \\ S_{4} S_{8} S_{10} \\ S_{3} S_{3} S_{11} S_{14} S_{15} \\ S_{3} S_{8} S_{4} S_{10} S_{10} S_{13} S_{12} \\ S_{3} S_{4} S_{8} S_{11} S_{11} S_{15} S_{14} \\ S_{3} S_{3} S_{10} S_{13} S_{12} \\ S_{8} S_{4} S_{11} \\ S_{3} \\ \end{array} \end{array}$

$\begin{array}{ccc} \begin{array}{c} S_{4} \\ S_{3} S_{14} \\ S_{8} S_{10} S_{12} \\ S_{3} S_{4} S_{11} S_{15} \\ S_{3} S_{4} S_{10} S_{13} \\ S_{8} S_{11} S_{14} \\ S_{3} S_{12} \\ S_{4} \\ \end{array} & \begin{array}{c} S_{5} \\ S_{2} S_{15} \\ S_{9} S_{10} S_{13} \\ S_{2} S_{5} S_{11} S_{14} \\ S_{2} S_{5} S_{10} S_{12} \\ S_{9} S_{11} S_{15} \\ S_{2} S_{13} \\ S_{5} \\ \end{array} & \begin{array}{c} S_{6} \\ S_{1} S_{15} \\ S_{7} S_{10} S_{13} \\ S_{1} S_{6} S_{11} S_{14} \\ S_{1} S_{6} S_{10} S_{12} \\ S_{7} S_{11} S_{15} \\ S_{1} S_{13} \\ S_{6} \\ \end{array} \end{array}$

$\begin{array}{ccc} \begin{array}{c} S_{7} \\ S_{1} S_{14} \\ S_{6} S_{10} S_{12} \\ S_{1} S_{7} S_{11} S_{15} \\ S_{1} S_{7} S_{10} S_{13} \\ S_{6} S_{11} S_{14} \\ S_{1} S_{12} \\ S_{7} \\ \end{array} & \begin{array}{c} S_{8} \\ S_{3} S_{15} \\ S_{4} S_{10} S_{13} \\ S_{3} S_{8} S_{11} S_{14} \\ S_{3} S_{8} S_{10} S_{12} \\ S_{4} S_{11} S_{15} \\ S_{3} S_{13} \\ S_{8} \\ \end{array} & \begin{array}{c} S_{9} \\ S_{2} S_{14} \\ S_{5} S_{10} S_{12} \\ S_{2} S_{9} S_{11} S_{15} \\ S_{2} S_{9} S_{10} S_{13} \\ S_{5} S_{11} S_{14} \\ S_{2} S_{12} \\ S_{9} \\ \end{array} \end{array}$

$\begin{array}{ccc} \begin{array}{c} S_{10} \\ S_{10} S_{11} S_{11} S_{14} S_{15} \\ S_{2} S_{3} S_{1} S_{11} S_{10} S_{10} S_{10} S_{15} S_{14} S_{12} S_{12} S_{13} S_{13} \\ S_{9} S_{4} S_{8} S_{7} S_{6} S_{5} S_{11} S_{11} S_{11} S_{10} S_{10} S_{10} S_{11} S_{14} S_{15} S_{13} S_{14} S_{15} S_{12} \\ S_{1} S_{3} S_{3} S_{2} S_{2} S_{1} S_{10} S_{10} S_{11} S_{11} S_{10} S_{12} S_{15} S_{13} S_{12} S_{14} S_{15} S_{13} S_{14} \\ S_{9} S_{6} S_{7} S_{5} S_{8} S_{4} S_{11} S_{10} S_{10} S_{10} S_{11} S_{14} S_{15} S_{12} S_{13} \\ S_{3} S_{1} S_{2} S_{11} S_{10} S_{15} S_{14} \\ S_{10} \\ \end{array} & \begin{array}{c} S_{11} \\ S_{2} S_{1} S_{3} S_{11} S_{10} S_{12} S_{13} \\ S_{8} S_{5} S_{4} S_{9} S_{7} S_{6} S_{11} S_{10} S_{11} S_{11} S_{10} S_{15} S_{13} S_{14} S_{12} \\ S_{1} S_{3} S_{2} S_{3} S_{1} S_{2} S_{11} S_{11} S_{10} S_{10} S_{11} S_{13} S_{14} S_{12} S_{12} S_{15} S_{13} S_{14} S_{15} \\ S_{5} S_{9} S_{6} S_{8} S_{7} S_{4} S_{11} S_{10} S_{10} S_{11} S_{10} S_{10} S_{11} S_{13} S_{12} S_{14} S_{15} S_{12} S_{13} \\ S_{2} S_{1} S_{3} S_{10} S_{11} S_{11} S_{11} S_{15} S_{15} S_{14} S_{14} S_{13} S_{12} \\ S_{11} S_{10} S_{10} S_{12} S_{13} \\ S_{11} \\ \end{array} & \begin{array}{c} S_{12} \\ S_{9} S_{4} S_{7} S_{11} S_{12} S_{14} \\ S_{1} S_{2} S_{3} S_{10} S_{11} S_{12} S_{14} S_{13} S_{14} \\ S_{6} S_{8} S_{5} S_{10} S_{11} S_{11} S_{10} S_{13} S_{12} S_{15} \\ S_{2} S_{1} S_{3} S_{11} S_{11} S_{10} S_{15} S_{15} S_{13} S_{12} \\ S_{7} S_{9} S_{4} S_{11} S_{10} S_{10} S_{12} S_{14} S_{13} \\ S_{11} S_{12} S_{14} S_{14} \\ S_{12} \\ \end{array} \end{array}$

$\begin{array}{ccc} \begin{array}{c} S_{13} \\ S_{8} S_{5} S_{6} S_{11} S_{13} S_{15} \\ S_{1} S_{2} S_{3} S_{10} S_{11} S_{12} S_{13} S_{15} S_{15} \\ S_{9} S_{7} S_{4} S_{11} S_{11} S_{10} S_{10} S_{14} S_{13} S_{12} \\ S_{1} S_{3} S_{2} S_{11} S_{10} S_{11} S_{13} S_{12} S_{14} S_{14} \\ S_{8} S_{6} S_{5} S_{11} S_{10} S_{10} S_{15} S_{13} S_{12} \\ S_{11} S_{15} S_{15} S_{13} \\ S_{13} \\ \end{array} & \begin{array}{c} S_{14} \\ S_{10} S_{14} S_{12} S_{12} \\ S_{4} S_{7} S_{9} S_{10} S_{11} S_{11} S_{15} S_{14} S_{12} \\ S_{3} S_{2} S_{1} S_{10} S_{11} S_{10} S_{14} S_{15} S_{13} S_{13} \\ S_{5} S_{8} S_{6} S_{10} S_{10} S_{11} S_{11} S_{13} S_{15} S_{14} \\ S_{3} S_{2} S_{1} S_{11} S_{10} S_{14} S_{15} S_{12} S_{12} \\ S_{4} S_{7} S_{9} S_{10} S_{12} S_{14} \\ S_{14} \\ \end{array} & \begin{array}{c} S_{15} \\ S_{10} S_{15} S_{13} S_{13} \\ S_{5} S_{6} S_{8} S_{10} S_{11} S_{11} S_{13} S_{14} S_{15} \\ S_{3} S_{2} S_{1} S_{10} S_{11} S_{10} S_{14} S_{15} S_{12} S_{12} \\ S_{4} S_{9} S_{7} S_{10} S_{10} S_{11} S_{11} S_{14} S_{15} S_{12} \\ S_{3} S_{2} S_{1} S_{10} S_{11} S_{15} S_{14} S_{13} S_{13} \\ S_{5} S_{6} S_{8} S_{10} S_{15} S_{13} \\ S_{15} \\ \end{array} \end{array}$

## Irreducible characters

All irreducible characters have height zero.

## Cartan matrix

$\left( \begin{array}{ccccccccccccccccccccc} 8 & 0 & 0 & 0 & 0 & 4 & 4 & 0 & 0 & 4 & 4 & 2 & 2 & 2 & 2 \\ 0 & 8 & 0 & 0 & 4 & 0 & 0 & 0 & 4 & 4 & 4 & 2 & 2 & 2 & 2 \\ 0 & 0 & 8 & 4 & 0 & 0 & 0 & 4 & 0 & 4 & 4 & 2 & 2 & 2 & 2 \\ 0 & 0 & 4 & 4 & 0 & 0 & 0 & 2 & 0 & 2 & 2 & 2 & 1 & 2 & 1 \\ 0 & 4 & 0 & 0 & 4 & 0 & 0 & 0 & 2 & 2 & 2 & 1 & 2 & 1 & 2 \\ 4 & 0 & 0 & 0 & 0 & 4 & 2 & 0 & 0 & 2 & 2 & 1 & 2 & 1 & 2 \\ 4 & 0 & 0 & 0 & 0 & 2 & 4 & 0 & 0 & 2 & 2 & 2 & 1 & 2 & 1 \\ 0 & 0 & 4 & 2 & 0 & 0 & 0 & 4 & 0 & 2 & 2 & 1 & 2 & 1 & 2 \\ 0 & 4 & 0 & 0 & 2 & 0 & 0 & 0 & 4 & 2 & 2 & 2 & 1 & 2 & 1 \\ 4 & 4 & 4 & 2 & 2 & 2 & 2 & 2 & 2 & 16 & 12 & 6 & 6 & 8 & 8 \\ 4 & 4 & 4 & 2 & 2 & 2 & 2 & 2 & 2 & 12 & 16 & 8 & 8 & 6 & 6 \\ 2 & 2 & 2 & 2 & 1 & 1 & 2 & 1 & 2 & 6 & 8 & 8 & 4 & 6 & 3 \\ 2 & 2 & 2 & 1 & 2 & 2 & 1 & 2 & 1 & 6 & 8 & 4 & 8 & 3 & 6 \\ 2 & 2 & 2 & 2 & 1 & 1 & 2 & 1 & 2 & 8 & 6 & 6 & 3 & 8 & 4 \\ 2 & 2 & 2 & 1 & 2 & 2 & 1 & 2 & 1 & 8 & 6 & 3 & 6 & 4 & 8 \end{array}\right)$

## Decomposition matrix

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