# M(16,14,6)

M(16,14,6) - $k(C_2 \times ((C_2)^3 : C_7))$
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Representative: $k((C_2)^4 : C_7)$ $(C_2)^4$ $C_5$ 16 7 1 $\left( \begin{array}{ccccccc} 4 & 2 & 2 & 2 & 2 & 2 & 2 \\ 2 & 4 & 2 & 2 & 2 & 2 & 2 \\ 2 & 2 & 4 & 2 & 2 & 2 & 2 \\ 2 & 2 & 2 & 4 & 2 & 2 & 2 \\ 2 & 2 & 2 & 2 & 4 & 2 & 2 \\ 2 & 2 & 2 & 2 & 2 & 4 & 2 \\ 2 & 2 & 2 & 2 & 2 & 2 & 4 \end{array} \right)$ Yes Yes Yes $\mathcal{O} (C_2 \times ((C_2)^3 : C_7))$ See below 1 No Yes M(16,14,7) Yes

## 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(16,14,6), then $B$ is in M(16,14,1), M(16,14,6) or M(16,14,13).

## Projective indecomposable modules

Labelling the simple $B$-modules by $S_1, S_2, S_3, S_4, S_5, S_6, S_7$, the projective indecomposable modules have Loewy structure as follows:

$\begin{array}{ccccccc} \begin{array}{c} S_1 \\ S_3 S_4 S_5 \\ S_2 S_6 S_7 \\ S_1 \\ \end{array} & \begin{array}{c} S_2 \\ S_1 S_4 S_7 \\ S_3 S_5 S_6 \\ S_2 \\ \end{array} & \begin{array}{c} S_3 \\ S_2 S_4 S_6 \\ S_1 S_5 S_7 \\ S_3 \\ \end{array} & \begin{array}{c} S_4 \\ S_5 S_6 S_7 \\ S_1 S_2 S_3 \\ S_4 \\ \end{array} & \begin{array}{c} S_5 \\ S_2 S_3 S_7 \\ S_1 S_4 S_6 \\ S_5 \\ \end{array} & \begin{array}{c} S_6 \\ S_1 S_2 S_5 \\ S_3 S_4 S_7 \\ S_6 \\ \end{array} & \begin{array}{c} S_7 \\ S_1 S_3 S_6 \\ S_2 S_4 S_5 \\ S_7 \\ \end{array} \end{array}$

## Irreducible characters

All irreducible characters have height zero.

## Decomposition matrix

$\left( \begin{array}{ccc} 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 \end{array}\right)$