# M(8,5,7)

M(8,5,7) - $B_0(kJ_1)$
Representative: $B_0(kJ_1)$ $C_2 \times C_2 \times C_2$ $C_7:C_3$ 8 5 1 $\left( \begin{array}{ccccc} 8 & 4 & 4 & 4 & 4 \\ 4 & 4 & 3 & 3 & 1 \\ 4 & 3 & 4 & 2 & 2 \\ 4 & 3 & 2 & 4 & 2 \\ 4 & 1 & 2 & 2 & 4 \\ \end{array} \right)$ Yes Yes Yes $B_0(\mathcal{O}J_1)$ $\left( \begin{array}{ccccc} 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 1 \\ 1 & 1 & 1 & 0 & 0 \\ 1 & 1 & 0 & 1 & 0 \\ 1 & 1 & 1 & 1 & 0 \\ 1 & 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 & 1 \\ 1 & 1 & 1 & 1 & 1 \\ \end{array}\right)$ 1 $1$[1] $(C_2 \wr S_4) \times C_2$[2] No Yes M(8,5,6), M(8,5,8) Yes M(8,5,7) (complete) M(8,5,7) (complete) M(16,14,14) (complete)

## Basic algebra

Quiver: a:<1,4>, b:<4,2>, c:<2,3>, d:<3,1>, e:<1,5>, f:<5,5>, g:<5,1>, h:<1,3>, i:<3,2>, j:<2,4>, k:<4,1>

Relations w.r.t. $k$:

The basic algebra for the block defined over $\mathcal{O}$ is described in [Ne02].

## Projective indecomposable modules

Labelling the simple $B$-modules by $1,2,3,4,5$, the projective indecomposable modules have Loewy structure as follows:

$\begin{array}{ccccc} \begin{array}{c} 1 \\ 3 \ 4 \ 5 \\ 1 \ 2 \ 1 \ 2 \ 1 \\ 3 \ 3 \ 4 \ 4 \ 5 \ 5 \\ 1 \ 2 \ 1 \ 2 \ 1 \\ 3 \ 4 \ 5 \\ 1 \\ \end{array}, & \begin{array}{c} 2 \\ 3 \ 4 \\ 1 \ 2 \ 1 \\ 3 \ 4 \ 5 \\ 1 \ 2 \ 1 \\ 3 \ 4 \\ 2 \\ \end{array}, & \begin{array}{c} 3 \\ 1 \ 2 \\ 3 \ 4 \ 5 \\ 1 \ 2 \ 1 \\ 3 \ 4 \ 5 \\ 1 \ 2 \\ 3 \\ \end{array} , & \begin{array}{c} 4 \\ 1 \ 2 \\ 3 \ 4 \ 5 \\ 1 \ 2 \ 1 \\ 3 \ 4 \ 5 \\ 1 \ 2 \\ 4 \\ \end{array}, & \begin{array}{ccc} 5 \\ 1 \ 5 \\ 3 \ 4 \\ 1 \ 2 \ 1 \\ 3 \ 4 \ 5 \\ 1 \\ 5 \\ \end{array} \end{array}$

## Irreducible characters

All irreducible characters have height zero.

## Notes

1. Shown by Eisele, using [Ne02].
2. See 8.7 of [Ru11], which uses GAP.