M(16,14,15)

M(16,14,15) - [math]B_0(k(C_2 \times \operatorname{Aut}(SL_2(8))))[/math]

[[File: |250px]]

Representative:	[math]B_0(k(C_2 \times \operatorname{Aut}(SL_2(8))))[/math]
Defect groups:	[math](C_2)^4[/math]
Inertial quotients:	[math]C_7:C_3[/math]
[math]k(B)=[/math]	16
[math]l(B)=[/math]	5
[math]{\rm mf}_k(B)=[/math]	1
[math]{\rm Pic}_k(B)=[/math]
Cartan matrix:	[math]\left( \begin{array}{ccccc} 8 & 4 & 4 & 8 & 4 \\ 4 & 8 & 4 & 8 & 4 \\ 4 & 4 & 8 & 8 & 4 \\ 8 & 8 & 8 & 16 & 6 \\ 4 & 4 & 4 & 6 & 4 \end{array}\right)[/math]
Defect group Morita invariant?	Yes
Inertial quotient Morita invariant?	Yes
[math]\mathcal{O}[/math]-Morita classes known?	Yes
[math]\mathcal{O}[/math]-Morita classes:	[math]B_0(\mathcal{O}(C_2 \times \operatorname{Aut}(SL_2(8))))[/math]
Decomposition matrices:	[math]\left( \begin{array}{ccccc} 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 & 0 \\ 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 1 & 0 \\ 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 2 & 1 \\ 1 & 1 & 1 & 2 & 1 \end{array}\right)[/math]
[math]{\rm mf}_\mathcal{O}(B)=[/math]	1
[math]{\rm Pic}_{\mathcal{O}}(B)=[/math]
[math]PI(B)=[/math]
Source algebras known?	No
Source algebra reps:
[math]k[/math]-derived equiv. classes known?	Yes
[math]k[/math]-derived equivalent to:	M(16,14,13), M(16,14,14)
[math]\mathcal{O}[/math]-derived equiv. classes known?	Yes
[math]p'[/math]-index covering blocks:
[math]p'[/math]-index covered blocks:
Index [math]p[/math] covering blocks:

Other notatable representatives

[math]B_0(k(C_2 \times {}^2G_2(q)))[/math], and the unique nonprincipal block with defect group $(C_2)^4$ of $C_2 \times \operatorname{Co}_3$.

Covering blocks and covered blocks

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

If [math]b[/math] is in M(16,14,15), then [math]B[/math] is in M(16,14,7) or M(16,14,15).

Projective indecomposable modules

Labelling the simple [math]B[/math]-modules by [math]S_1, S_2, S_3, S_4, S_5[/math], the projective indecomposable modules have Loewy structure as follows:

[math]\begin{array}{ccccc} \begin{array}{c} S_1 \\ S_1 S_4 \\ S_1 S_2 S_3 S_4 S_5 \\ S_1 S_2 S_3 S_4 S_4 S_5 \\ S_1 S_2 S_3 S_4 S_4 S_5 \\ S_1 S_2 S_3 S_4 S_5 \\ S_1 S_4 \\ S_1 \\ \end{array} & \begin{array}{c} S_2 \\ S_2 S_4 \\ S_1 S_2 S_3 S_4 S_5 \\ S_1 S_2 S_3 S_4 S_4 S_5 \\ S_1 S_2 S_3 S_4 S_4 S_5 \\ S_1 S_2 S_3 S_4 S_5 \\ S_2 S_4 \\ S_2 \\ \end{array} & \begin{array}{c} S_3 \\ S_3 S_4 \\ S_1 S_2 S_3 S_4 S_5 \\ S_1 S_2 S_3 S_4 S_4 S_5 \\ S_1 S_2 S_3 S_4 S_4 S_5 \\ S_1 S_2 S_3 S_4 S_5 \\ S_3 S_4 \\ S_3 \\ \end{array} & \begin{array}{c} S_4 \\ S_1 S_2 S_3 S_4 S_5 \\ S_1 S_2 S_3 S_4 S_4 S_4 S_5 \\ S_1 S_1 S_2 S_2 S_3 S_3 S_4 S_4 S_4 S_5 \\ S_1 S_1 S_2 S_2 S_3 S_3 S_4 S_4 S_4 S_5 \\ S_1 S_2 S_3 S_4 S_4 S_4 S_5 \\ S_1 S_2 S_3 S_4 S_5 \\ S_4 \\ \end{array} & \begin{array}{c} S_5 \\ S_4 S_5 \\ S_1 S_2 S_3 S_4 \\ S_1 S_2 S_3 S_4 \\ S_1 S_2 S_3 S_4 \\ S_1 S_2 S_3 S_4 \\ S_4 S_5 \\ S_5 \\ \end{array} \end{array} [/math]

Irreducible characters

All irreducible characters have height zero.

Back to [math](C_2)^4[/math]