# M(8,5,1)

M(8,5,1) - $k(C_2 \times C_2 \times C_2)$
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Representative: $k(C_2 \times C_2 \times C_2)$ $C_2 \times C_2 \times C_2$ $1$ 8 1 1 $\left( \begin{array}{c} 8 \\ \end{array} \right)$ Yes Yes Yes $\mathcal{O} (C_2 \times C_2 \times C_2)$ $\left( \begin{array}{c} 1 \\ 1 \\ \vdots \\ 1 \\ \end{array}\right)$ 1 $(C_2 \times C_2 \times C_2):GL_3(2)$ {{{PIgroup}}} No Yes Forms a derived equivalence class Yes

These are nilpotent blocks.

## Basic algebra

Quiver: a:<1,1>, b:<1,1>, c:<1,1>

Relations w.r.t. $k$: a^2=b^2=b^c=0, ab+ba=ac+ca=bc+cb=0

## Covering blocks and covered blocks

Let $N \triangleleft G$ with $p'$-index and let $B$ be a block of $\mathcal{O} G$ covering a block $b$ of $\mathcal{O} N$.

If $b$ is in M(8,5,1), then $B$ is in M(8,5,1), M(8,5,3), M(8,5,4) or M(8,5,6).

## Projective indecomposable modules

Labelling the unique simple $B$-module by $S_1$, the unique projective indecomposable module has Loewy structure as follows:

$\begin{array}{c} S_1 \\ S_1 S_1 S_1 S_1 \\ S_1 S_1 S_1 S_1 S_1 S_1 \\ S_1 S_1 S_1 S_1 \\ S_1 \\ \end{array}$

## Irreducible characters

All irreducible characters have height zero.