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M(16,14,16) - [math]b_2(k((C_2)^4:3^{1+2}_+))[/math]
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Representative: [math]b_2(k((C_2)^4:3^{1+2}_+))[/math]
Defect groups: [math](C_2)^4[/math]
Inertial quotients: [math]C_3 \times C_3[/math]
[math]k(B)=[/math] 8
[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]b_2(\mathcal{O}((C_2)^4:3^{1+2}_+))[/math]
Decomposition matrices: [math]\left( \begin{array}{c} 1 \\ 1 \\ \vdots \\ 1 \\ 3 \end{array}\right)[/math]
[math]{\rm mf}_\mathcal{O}(B)=[/math] 1
[math]{\rm Pic}_{\mathcal{O}}(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.

Basic algebra

Other notatable representatives

Any nonprincipal block of [math](C_2)^4:3^{1+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(16,14,16), then [math]B[/math] is in M(16,14,3) or M(16,14,16).

Projective indecomposable modules

Labelling the unique simple [math]B[/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 \\ \end{array} [/math]

Irreducible characters

All irreducible characters have height zero.

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