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M(16,10,1) - [math]k(C_4 \times C_2 \times C_2)[/math]
Representative: [math]k(C_4 \times C_2 \times C_2)[/math]
Defect groups: [math]C_4 \times C_2 \times C_2[/math]
Inertial quotients: [math]1[/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} 16 \\ \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_2 \times C_2)[/math]
Decomposition matrices: [math]\left( \begin{array}{c} 1 \\ 1 \\ \vdots \\ 1 \\ \end{array}\right)[/math]
[math]{\rm mf}_\mathcal{O}(B)=[/math] 1
[math]{\rm Pic}_{\mathcal{O}}(B)=[/math] [math]\mathcal{L}(B)=(C_4 \times C_2 \times C_2):{\rm Aut}(C_4 \times C_2 \times C_2)[/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: M(16,10,1), M(16,10,3) (complete)
[math]p'[/math]-index covered blocks: M(16,10,1), M(16,10,3)[1] (complete)
Index [math]p[/math] covering blocks:

These are nilpotent blocks.

Basic algebra

Quiver: a:<1,1>, b:<1,1>, c:<1,1>

Relations w.r.t. [math]k[/math]: [math]a^4=b^2=c^2=0[/math], [math]ab+ba=ac+ca=bc+cb=0[/math]

Projective indecomposable modules

Irreducible characters

All irreducible characters have height zero.

Back to [math]C_4 \times C_2 \times C_2[/math]


  1. For example consider the block of [math]C_4 \times PSL_3(7)[/math] covering block number 2 of [math]PSL_3(7)[/math] in the labelling used in [1]. We have [math]C_4 \times PSL_3(7) \triangleleft C_4 \times PGL_3(7)[/math].