M(16,14,6)

M(16,14,6) - $k(C_2 \times ((C_2)^3 : C_7))$

[[File: |250px]]

Representative:	$k((C_2)^4 : C_7)$
Defect groups:	$(C_2)^4$
Inertial quotients:	$C_5$
$k(B)=$	16
$l(B)=$	7
${\rm mf}_k(B)=$	1
${\rm Pic}_k(B)=$
Cartan matrix:	$\left( \begin{array}{ccccccc} 4 & 2 & 2 & 2 & 2 & 2 & 2 \\ 2 & 4 & 2 & 2 & 2 & 2 & 2 \\ 2 & 2 & 4 & 2 & 2 & 2 & 2 \\ 2 & 2 & 2 & 4 & 2 & 2 & 2 \\ 2 & 2 & 2 & 2 & 4 & 2 & 2 \\ 2 & 2 & 2 & 2 & 2 & 4 & 2 \\ 2 & 2 & 2 & 2 & 2 & 2 & 4 \end{array} \right)$
Defect group Morita invariant?	Yes
Inertial quotient Morita invariant?	Yes
$\mathcal{O}$-Morita classes known?	Yes
$\mathcal{O}$-Morita classes:	$\mathcal{O} (C_2 \times ((C_2)^3 : C_7))$
Decomposition matrices:	See below
${\rm mf}_\mathcal{O}(B)=$	1
${\rm Pic}_{\mathcal{O}}(B)=$
$PI(B)=$
Source algebras known?	No
Source algebra reps:
$k$-derived equiv. classes known?	Yes
$k$-derived equivalent to:	M(16,14,7)
$\mathcal{O}$-derived equiv. classes known?	Yes
$p'$-index covering blocks:
$p'$-index covered blocks:
Index $p$ covering blocks:

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

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

Back to $(C_2)^4$