# M(8,5,3)

M(8,5,3) - $k(A_4 \times C_2)$
Representative: $k(A_4 \times C_2)$ $C_2 \times C_2 \times C_2$ $C_3$ 8 3 1 $\left( \begin{array}{ccc} 4 & 2 & 2 \\ 2 & 4 & 2 \\ 2 & 2 & 4 \\ \end{array} \right)$ Yes Yes Yes $\mathcal{O} (A_4 \times C_2)$ $\left( \begin{array}{ccc} 1 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \\ \end{array}\right)$ 1 $S_3 \times C_2$[1] $S_4 \times C_2 \times C_2$[2] No Yes M(8,5,2) Yes M(16,10,1)[3], M(8,5,3) (complete) M(16,10,3), M(16,14,3)

## Basic algebra

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

Relations w.r.t. $k$: $k$: $ab=bc=ca=0$, $df=fe=ed=0$, $ad=fc$, $be=da$, $cf=eb$, $g^2=h^2=i^2=0$, $ah=ga$, $bi=hb$, $cg=ic$, $dg=hd$, $eh=ie$, $fi=gf$

## Projective indecomposable modules

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

$\begin{array}{ccc} \begin{array}{ccc} & S_1 & \\ S_1 & S_2 & S_3 \\ S_2 & S_3 & S_1 \\ & S_1 & \\ \end{array}, & \begin{array}{ccc} & S_2 & \\ S_1 & S_3 & S_2 \\ S_2 & S_1 & S_3 \\ & S_2 & \\ \end{array}, & \begin{array}{ccc} & S_3 & \\ S_1 & S_2 & S_3 \\ S_3 & S_1 & S_2 \\ & S_3 & \\ \end{array} \end{array}$

## Irreducible characters

All irreducible characters have height zero.

## Notes

1. See [EL18c]
2. By Theorem 3.7 of [EL18c].
3. For example consider the block of $PSL_3(7) \times C_2$ covering the block number 2 of $PSL_3(7) \triangleleft PGL_3(7)$ in the labelling used in [1]. The covering block of $PGL_3(7) \times C_2$ is nilpotent.