# M(16,14,8)

M(16,14,8) - $k(A_4 \times A_4)$
[[File: |250px]]
Representative: $k(A_4 \times A_4)$ $(C_2)^4$ $C_3 \times C_3$ 16 9 1 $\left( \begin{array}{ccccccccc} 4 & 2 & 2 & 1 & 1 & 1 & 1 & 2 & 2 \\ 2 & 4 & 1 & 2 & 1 & 2 & 1 & 1 & 2 \\ 2 & 1 & 4 & 2 & 2 & 1 & 1 & 2 & 1 \\ 1 & 2 & 2 & 4 & 2 & 2 & 1 & 1 & 1 \\ 1 & 1 & 2 & 2 & 4 & 1 & 2 & 1 & 2 \\ 1 & 2 & 1 & 2 & 1 & 4 & 2 & 2 & 1 \\ 1 & 1 & 1 & 1 & 2 & 2 & 4 & 2 & 2 \\ 2 & 1 & 2 & 1 & 1 & 2 & 2 & 4 & 1 \\ 2 & 2 & 1 & 1 & 2 & 1 & 2 & 1 & 4 \end{array} \right)$ Yes Yes Yes $\mathcal{O} (A_4 \times A_4)$ See below 1 $S_3 \wr C_2$ No Yes M(16,14,9), M(16,14,10) Yes

## Covering blocks and covered blocks

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

If $b$ is in M(16,14,8), then $B$ is in M(16,14,3), M(16,14,4), M(16,14,8).

## Projective indecomposable modules

Labelling the simple $B$-modules by $S_1, \dots, S_9$, the projective indecomposable modules have Loewy structure as follows:

$\begin{array}{ccccccccc} \begin{array}{c} S_1 \\ S_2 S_3 S_8 S_9 \\ S_1 S_1 S_4 S_5 S_6 S_7 \\ S_2 S_3 S_8 S_9 \\ S_1 \\ \end{array} & \begin{array}{c} S_2 \\ S_1 S_4 S_6 S_9 \\ S_2 S_2 S_3 S_5 S_7 S_8 \\ S_1 S_4 S_6 S_9 \\ S_2 \\ \end{array} & \begin{array}{c} S_3 \\ S_1 S_4 S_5 S_8 \\ S_2 S_3 S_3 S_6 S_7 S_9 \\ S_1 S_4 S_5 S_8 \\ S_3 \\ \end{array} & \begin{array}{c} S_4 \\ S_2 S_3 S_5 S_6 \\ S_1 S_4 S_4 S_7 S_8 S_9 \\ S_2 S_3 S_5 S_6 \\ S_4 \\ \end{array} & \begin{array}{c} S_5 \\ S_3 S_4 S_7 S_9 \\ S_1 S_2 S_5 S_5 S_6 S_8 \\ S_3 S_4 S_7 S_9 \\ S_5 \\ \end{array} & \begin{array}{c} S_6 \\ S_2 S_4 S_7 S_8 \\ S_1 S_3 S_5 S_6 S_6 S_9 \\ S_2 S_4 S_7 S_8 \\ S_6 \\ \end{array} & \begin{array}{c} S_7 \\ S_5 S_6 S_8 S_9 \\ S_1 S_2 S_3 S_4 S_7 S_7 \\ S_5 S_6 S_8 S_9 \\ S_7 \\ \end{array} & \begin{array}{c} S_8 \\ S_1 S_3 S_6 S_7 \\ S_2 S_4 S_5 S_8 S_8 S_9 \\ S_1 S_3 S_6 S_7 \\ S_8 \\ \end{array} & \begin{array}{c} S_9 \\ S_1 S_2 S_5 S_7 \\ S_3 S_4 S_6 S_8 S_9 S_9 \\ S_1 S_2 S_5 S_7 \\ S_9 \\ \end{array} \end{array}$

## Irreducible characters

All irreducible characters have height zero.

## Decomposition matrix

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