M(32,51,32)

M(32,51,32) - [math]b_2(k((C_2)^4:3^{1+2}_+) \times C_2)[/math]

[[File: |250px]]

Representative:	[math]b_2(k((C_2)^4:3^{1+2}_+) \times C_2)[/math]
Defect groups:	[math](C_2)^5[/math]
Inertial quotients:	[math]C_3 \times C_3[/math]
[math]k(B)=[/math]	16
[math]l(B)=[/math]	1
[math]{\rm mf}_k(B)=[/math]	1
[math]{\rm Pic}_k(B)=[/math]
Cartan matrix:	[math]\left( \begin{array}{c} 32 \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_2(\mathcal{O}((C_2)^4:3^{1+2}_+) \times C_2)[/math]
Decomposition matrices:	[math]\left( \begin{array}{c} 1 \\ 1 \\ \vdots \\ 1 \\ 3 \\ 3 \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:	Forms a derived equivalence class
[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:

This Morita equivalence class contains only non-principal blocks.

Other notatable representatives

Any nonprincipal block of [math](C_2)^4:3^{1+2}_- \times C_2[/math].

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(32,51,32), then [math]B[/math] is in M(32,51,2) or M(32,51,32).

Projective indecomposable modules

Labelling the unique simple [math]b_2[/math]-module by [math]S_1[/math], the unique projective indecomposable module has Loewy structure as follows:

[math]\begin{array}{c} S_{1} \\ S_{1} S_{1} S_{1} S_{1} S_{1} \\ S_{1} S_{1} S_{1} S_{1} S_{1} S_{1} S_{1} S_{1} S_{1} S_{1} \\ S_{1} S_{1} S_{1} S_{1} S_{1} S_{1} S_{1} S_{1} S_{1} S_{1} \\ S_{1} S_{1} S_{1} S_{1} S_{1} \\ S_{1} \\ \end{array} [/math]

Irreducible characters

All irreducible characters have height zero.

Back to [math](C_2)^5[/math]