# M(32,51,34)

M(32,51,34) - $b_2(k((SL_2(8)\times (C_2)^2):3^{1+2}_+))$
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Representative: $b_2(k((SL_2(8)\times (C_2)^2):3^{1+2}_+))$ $(C_2)^5$ $C_7:C_3 \times C_3$ 16 7 1 $\left( \begin{array}{ccccccc} 32 & 16 & 16 & 16 & 8 & 8 & 8 \\ 16 & 12 & 10 & 10 & 4 & 5 & 3 \\ 16 & 10 & 12 & 10 & 3 & 4 & 5 \\ 16 & 10 & 10 & 12 & 5 & 3 & 4 \\ 8 & 4 & 3 & 5 & 4 & 2 & 2 \\ 8 & 5 & 4 & 3 & 2 & 4 & 2 \\ 8 & 3 & 5 & 4 & 2 & 2 & 4 \end{array} \right)$ Yes Yes Yes $b_2(\mathcal{O}((SL_2(8)\times (C_2)^2):3^{1+2}_+))$ $\left( \begin{array}{ccccccc} 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 1 & 0 & 0 \\ 1 & 0 & 1 & 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 1 & 0 & 0 & 1 \\ 1 & 1 & 1 & 0 & 1 & 0 & 0 \\ 1 & 0 & 1 & 1 & 0 & 1 & 0 \\ 1 & 1 & 0 & 1 & 0 & 0 & 1 \\ 3 & 1 & 1 & 1 & 1 & 1 & 1 \\ 3 & 2 & 2 & 2 & 1 & 1 & 1 \end{array}\right)$ 1 No No No

This Morita equivalence class contains only non-principal blocks.

It is unknown whether this class is derived equivalent to M(32,51,33); if not, it forms its own derived equivalence class.

## Other notatable representatives

Any nonprincipal block with defect group $(C_2)^5$ of $(SL_2(8)\times (C_2)^2):3^{1+2}_-$.

## 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(32,51,34), then $B$ is in M(32,51,15), M(32,51,19) or M(32,51,34).

## Projective indecomposable modules

Labelling the simple $b_2$-modules by $S_1,\dots, S_7$, the projective indecomposable modules have Loewy structure as follows:

$\begin{array}{cccc} \begin{array}{c} S_{1} \\ S_{1} S_{1} S_{3} S_{4} S_{2} \\ S_{1} S_{1} S_{1} S_{1} S_{2} S_{4} S_{3} S_{4} S_{3} S_{2} S_{6} S_{5} S_{7} \\ S_{1} S_{1} S_{1} S_{1} S_{1} S_{1} S_{3} S_{2} S_{3} S_{2} S_{2} S_{3} S_{4} S_{4} S_{4} S_{5} S_{7} S_{6} S_{5} S_{7} S_{6} \\ S_{1} S_{1} S_{1} S_{1} S_{1} S_{1} S_{2} S_{4} S_{2} S_{3} S_{4} S_{3} S_{3} S_{2} S_{2} S_{4} S_{4} S_{3} S_{6} S_{7} S_{6} S_{5} S_{5} S_{7} \\ S_{1} S_{1} S_{1} S_{1} S_{1} S_{1} S_{2} S_{4} S_{4} S_{2} S_{3} S_{3} S_{4} S_{3} S_{2} S_{5} S_{5} S_{7} S_{6} S_{6} S_{7} \\ S_{1} S_{1} S_{1} S_{1} S_{2} S_{3} S_{3} S_{4} S_{4} S_{2} S_{6} S_{7} S_{5} \\ S_{1} S_{1} S_{3} S_{4} S_{2} \\ S_{1} \\ \end{array} & \begin{array}{c} S_{2} \\ S_{1} S_{3} S_{4} S_{6} \\ S_{1} S_{1} S_{3} S_{2} S_{4} S_{2} S_{5} S_{7} \\ S_{1} S_{1} S_{1} S_{3} S_{4} S_{2} S_{2} S_{3} S_{4} S_{5} S_{6} \\ S_{1} S_{1} S_{1} S_{1} S_{3} S_{2} S_{4} S_{3} S_{4} S_{2} S_{6} S_{7} \\ S_{1} S_{1} S_{1} S_{2} S_{3} S_{4} S_{3} S_{2} S_{4} S_{6} S_{5} \\ S_{1} S_{1} S_{4} S_{2} S_{3} S_{2} S_{5} S_{7} \\ S_{1} S_{3} S_{4} S_{6} \\ S_{2} \\ \end{array} & \begin{array}{c} S_{3} \\ S_{1} S_{2} S_{4} S_{7} \\ S_{1} S_{1} S_{3} S_{2} S_{3} S_{4} S_{5} S_{6} \\ S_{1} S_{1} S_{1} S_{3} S_{4} S_{4} S_{3} S_{2} S_{2} S_{7} S_{6} \\ S_{1} S_{1} S_{1} S_{1} S_{4} S_{2} S_{3} S_{4} S_{2} S_{3} S_{7} S_{5} \\ S_{1} S_{1} S_{1} S_{4} S_{3} S_{4} S_{3} S_{2} S_{2} S_{7} S_{6} \\ S_{1} S_{1} S_{4} S_{3} S_{2} S_{3} S_{6} S_{5} \\ S_{1} S_{4} S_{2} S_{7} \\ S_{3} \\ \end{array} \end{array}$

$\begin{array}{ccc} \begin{array}{c} S_{4} \\ S_{1} S_{2} S_{3} S_{5} \\ S_{1} S_{1} S_{2} S_{4} S_{3} S_{4} S_{7} S_{6} \\ S_{1} S_{1} S_{1} S_{4} S_{3} S_{2} S_{4} S_{3} S_{2} S_{5} S_{7} \\ S_{1} S_{1} S_{1} S_{1} S_{3} S_{2} S_{4} S_{3} S_{4} S_{2} S_{6} S_{5} \\ S_{1} S_{1} S_{1} S_{2} S_{3} S_{4} S_{3} S_{2} S_{4} S_{5} S_{7} \\ S_{1} S_{1} S_{4} S_{3} S_{4} S_{2} S_{7} S_{6} \\ S_{1} S_{2} S_{3} S_{5} \\ S_{4} \\ \end{array} & \begin{array}{c} S_{5} \\ S_{4} S_{7} S_{6} \\ S_{1} S_{3} S_{2} S_{5} \\ S_{1} S_{1} S_{2} S_{4} \\ S_{1} S_{1} S_{4} S_{3} \\ S_{1} S_{1} S_{2} S_{4} \\ S_{1} S_{2} S_{3} S_{5} \\ S_{4} S_{7} S_{6} \\ S_{5} \\ \end{array} & \begin{array}{c} S_{6} \\ S_{2} S_{5} S_{7} \\ S_{1} S_{4} S_{3} S_{6} \\ S_{1} S_{1} S_{3} S_{2} \\ S_{1} S_{1} S_{4} S_{2} \\ S_{1} S_{1} S_{3} S_{2} \\ S_{1} S_{4} S_{3} S_{6} \\ S_{2} S_{7} S_{5} \\ S_{6} \\ \end{array} & \begin{array}{c} S_{7} \\ S_{3} S_{6} S_{5} \\ S_{1} S_{4} S_{2} S_{7} \\ S_{1} S_{1} S_{4} S_{3} \\ S_{1} S_{1} S_{3} S_{2} \\ S_{1} S_{1} S_{4} S_{3} \\ S_{1} S_{2} S_{4} S_{7} \\ S_{3} S_{5} S_{6} \\ S_{7} \\ \end{array} \end{array}$

## Irreducible characters

All irreducible characters have height zero.