M(16,14,3)

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
M(16,14,3) - $k(C_2 \times C_2 \times A_4)$
[[File: |250px]]
Representative: $k(C_2 \times C_2 \times A_4)$ $(C_2)^4$ $C_3$ 16 3 1 $\left( \begin{array}{ccc} 8 & 4 & 4\\ 4 & 8 & 4 \\ 4 & 4 & 8 \end{array} \right)$ Yes Yes Yes $\mathcal{O} (C_2 \times C_2 \times A_4)$ See below 1 $(C_2 \times C_2):S_3 \times S_3$ No Yes M(16,14,2) 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(16,14,3), then $B$ is in M(16,14,1), M(16,14,3), M(16,14,8), M(16,14,16).

Projective indecomposable modules

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

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

Irreducible characters

All irreducible characters have height zero.

Decomposition matrix

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