# M(16,14,4)

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M(16,14,4) - $k((C_2)^4 : C_3)$
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Representative: $k((C_2)^4 : C_3)$ $(C_2)^4$ $C_3$ 8 3 1 $\left( \begin{array}{ccc} 6 & 5 & 5\\ 5 & 6 & 5 \\ 5 & 5 & 6 \end{array} \right)$ Yes Yes Yes $\mathcal{O} ((C_2)^4 : C_3)$ $\left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{array}\right)$ 1 No Yes Forms a derived equivalence class Yes

The action of $C_3$ on the defect group, distinguished from the one in M(16,4,3), comes from the 5th power of a Singer cycle.

## 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(16,14,4), then $B$ is in M(16,14,1), M(16,14,4), M(16,14,8) or M(16,14,11).

## Projective indecomposable modules

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

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

## Irreducible characters

All irreducible characters have height zero.