M(16,2,2)

M(9,2,3) - [math]k((C_4 \times C_4):C_3)[/math]

Representative:	[math]k((C_4 \times C_4):C_3)[/math]
Defect groups:	[math]C_4 \times C_4[/math]
Inertial quotients:	[math]C_3[/math]
[math]k(B)=[/math]	8
[math]l(B)=[/math]	3
[math]{\rm mf}_k(B)=[/math]	1
[math]{\rm Pic}_k(B)=[/math]	[math][/math]
Cartan matrix:	[math]\left( \begin{array}{ccc} 6 & 5 & 5 \\ 5 & 6 & 5 \\ 5 & 5 & 6 \\ \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]\mathcal{O}((C_4 \times C_4):C_3)[/math]
Decomposition matrices:	[math]\left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \\ \end{array}\right)[/math]
[math]{\rm mf}_\mathcal{O}(B)=[/math]	1
[math]{\rm Pic}_{\mathcal{O}}(B)=[/math]	[math]S_3[/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 its own derived equivalence class
[math]\mathcal{O}[/math]-derived equiv. classes known?	Yes
[math]p'[/math]-index covering blocks:
[math]p'[/math]-index covered blocks:	M(16,2,1)
Index [math]p[/math] covering blocks:

Basic algebra

Quiver: a:<1,2>, b:<2,3>, c:<3,1>, d:<2,1>, e:<3,2>, f:<1,3>

Relations w.r.t. [math]k[/math]: abca=bcab=cabc=0, dfed=fedf=edfe=0, ad=fc, be=da, cf=eb

Projective indecomposable modules

Labelling the simple [math]B[/math]-modules by [math]1,2,3[/math], the projective indecomposable modules have Loewy structure as follows:

[math]\begin{array}{ccc} \begin{array}{c} 1 \\ 2 \ 3 \\ 3 \ 1 \ 2 \\ 1 \ 2 \ 3 \ 1 \\ 3 \ 1 \ 2 \\ 2 \ 3 \\ 1 \\ \end{array}, & \begin{array}{c} 2 \\ 1 \ 3 \\ 3 \ 2 \ 1 \\ 2 \ 1 \ 3 \ 2 \\ 3 \ 2 \ 1 \\ 1 \ 3 \\ 2 \\ \end{array}, & \begin{array}{c} 3 \\ 1 \ 2 \\ 2 \ 3 \ 1 \\ 3 \ 1 \ 2 \ 3 \\ 2 \ 3 \ 1 \\ 1 \ 2 \\ 2 \\ \end{array} \end{array} [/math]

Irreducible characters

All irreducible characters have height zero.

Back to [math]C_4 \times C_4[/math]