# M(8,5,6)

M(8,5,6) - $k((C_2 \times C_2 \times C_2):(C_7:C_3))$
Representative: $k((C_2 \times C_2 \times C_2):(C_7:C_3))$ $C_2 \times C_2 \times C_2$ $C_7:C_3$ 8 5 1 $\left( \begin{array}{ccccccc} 2 & 0 & 0 & 1 & 1 \\ 0 & 2 & 0 & 1 & 1 \\ 0 & 0 & 2 & 1 & 1 \\ 1 & 1 & 1 & 4 & 3 \\ 1 & 1 & 1 & 3 & 4 \\ \end{array} \right)$ Yes Yes Yes $\mathcal{O} ((C_2 \times C_2 \times C_2):(C_7:C_3))$ $\left( \begin{array}{ccccc} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 1 & 1 \\ 0 & 1 & 0 & 1 & 1 \\ 0 & 0 & 1 & 1 & 1 \\ \end{array}\right)$ 1 $C_3$[1] {{{PIgroup}}} No Yes M(8,5,7), M(8,5,8) Yes {{{pcoveringblocks}}}

## Basic algebra

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

Relations w.r.t. $k$:

## Projective indecomposable modules

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

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

## Irreducible characters

All irreducible characters have height zero.

1. See [EL18c]