# M(32,51,28)

M(32,51,28) - $B_0(k({\rm Aut}(SL_2(8)) \times A_4))$
[[File: |250px]]
Representative: $B_0(k({\rm Aut}(SL_2(8)) \times A_4))$ $(C_2)^5$ $(C_{7}:C_3) \times C_3$ 32 15 1 See below. Yes Yes Yes $B_0(\mathcal{O}({\rm Aut}(SL_2(8)) \times A_4))$ See below. 1 No Yes M(32,51,24), M(32,51,25), M(32,51,26), M(32,51,27), M(32,51,29) 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(32,51,28), then $B$ is in M(32,51,7), M(32,51,15), M(32,51,21) or M(32,51,28).

## Irreducible characters

All irreducible characters have height zero.

## Cartan matrix

$\left( \begin{array}{ccccccccccccccccccccc} 8 & 4 & 4 & 4 & 4 & 2 & 2 & 2 & 2 & 8 & 4 & 4 & 4 & 2 & 2\\ 4 & 8 & 4 & 2 & 2 & 2 & 4 & 2 & 4 & 8 & 4 & 4 & 4 & 2 & 2\\ 4 & 4 & 8 & 2 & 2 & 4 & 2 & 4 & 2 & 8 & 4 & 4 & 4 & 2 & 2\\ 4 & 2 & 2 & 8 & 4 & 2 & 2 & 4 & 4 & 4 & 8 & 4 & 2 & 4 & 2\\ 4 & 2 & 2 & 4 & 8 & 4 & 4 & 2 & 2 & 4 & 4 & 8 & 2 & 2 & 4\\ 2 & 2 & 4 & 2 & 4 & 8 & 4 & 4 & 2 & 4 & 4 & 8 & 2 & 2 & 4\\ 2 & 4 & 2 & 2 & 4 & 4 & 8 & 2 & 4 & 4 & 4 & 8 & 2 & 2 & 4\\ 2 & 2 & 4 & 4 & 2 & 4 & 2 & 8 & 4 & 4 & 8 & 4 & 2 & 4 & 2\\ 2 & 4 & 2 & 4 & 2 & 2 & 4 & 4 & 8 & 4 & 8 & 4 & 2 & 4 & 2\\ 8 & 8 & 8 & 4 & 4 & 4 & 4 & 4 & 4 & 16 & 8 & 8 & 6 & 3 & 3\\ 4 & 4 & 4 & 8 & 4 & 4 & 4 & 8 & 8 & 8 & 16 & 8 & 3 & 6 & 3\\ 4 & 4 & 4 & 4 & 8 & 8 & 8 & 4 & 4 & 8 & 8 & 16 & 3 & 3 & 6\\ 4 & 4 & 4 & 2 & 2 & 2 & 2 & 2 & 2 & 6 & 3 & 3 & 4 & 2 & 2\\ 2 & 2 & 2 & 4 & 2 & 2 & 2 & 4 & 4 & 3 & 6 & 3 & 2 & 4 & 2\\ 2 & 2 & 2 & 2 & 4 & 4 & 4 & 2 & 2 & 3 & 3 & 6 & 2 & 2 & 4 \end{array}\right)$

## Decomposition matrix

$\left( \begin{array}{ccccccccccccccccccccc} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 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 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 1 & 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 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 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 & 1 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 1 & 1 & 1 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 2 & 0 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 2 & 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 2 & 0 & 0 & 1 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 2 & 2 & 2 & 1 & 1 & 1 \end{array}\right)$