M(32,51,33)

M(32,51,32) - [math]b_2(k((C_2)^5:(C_7:3^{1+2}_+)))[/math]

[[File: |250px]]

Representative:	[math]b_2(k((C_2)^5:(C_7:3^{1+2}_+)))[/math]
Defect groups:	[math](C_2)^5[/math]
Inertial quotients:	[math]C_7:C_3 \times C_3[/math]
[math]k(B)=[/math]	16
[math]l(B)=[/math]	7
[math]{\rm mf}_k(B)=[/math]	1
[math]{\rm Pic}_k(B)=[/math]
Cartan matrix:	[math]\left( \begin{array}{ccccccc} 8 & 4 & 4 & 4 & 4 & 4 & 4 \\ 4 & 6 & 5 & 5 & 4 & 4 & 4 \\ 4 & 5 & 6 & 5 & 4 & 4 & 4 \\ 4 & 5 & 5 & 6 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 & 6 & 5 & 5 \\ 4 & 4 & 4 & 4 & 5 & 6 & 5 \\ 4 & 4 & 4 & 4 & 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]b_2(k((C_2)^5:(C_7:3^{1+2}_+)))[/math]
Decomposition matrices:	[math]\left( \begin{array}{ccccccc} 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 & 1 \\ 0 & 1 & 1 & 1 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 & 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]PI(B)=[/math]
Source algebras known?	No
Source algebra reps:
[math]k[/math]-derived equiv. classes known?	Yes
[math]k[/math]-derived equivalent to:
[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)^5:(C_7:3^{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(32,51,33), then [math]B[/math] is in M(32,51,13), M(32,51,17) or M(32,51,33).

Projective indecomposable modules

Labelling the simple [math]b_2[/math]-modules by [math]S_1,\dots, S_7 [/math], the projective indecomposable modules have Loewy structure as follows:

[math]\begin{array}{cccc} \begin{array}{c} S_{1} \\ S_{1} S_{4} S_{1} S_{6} S_{7} \\ S_{2} S_{7} S_{4} S_{3} S_{4} S_{6} S_{6} S_{5} S_{7} S_{1} \\ S_{5} S_{4} S_{2} S_{6} S_{3} S_{2} S_{3} S_{1} S_{7} S_{5} \\ S_{1} S_{2} S_{5} S_{3} S_{1} \\ S_{1} \\ \end{array} & \begin{array}{c} S_{2} \\ S_{1} S_{3} S_{6} S_{5} S_{3} \\ S_{7} S_{1} S_{2} S_{2} S_{1} S_{7} S_{5} S_{5} S_{4} S_{4} \\ S_{2} S_{6} S_{6} S_{7} S_{1} S_{3} S_{6} S_{3} S_{2} S_{4} \\ S_{7} S_{3} S_{5} S_{5} S_{4} \\ S_{2} \\ \end{array} & \begin{array}{c} S_{3} \\ S_{5} S_{2} S_{1} S_{5} S_{7} \\ S_{3} S_{4} S_{2} S_{3} S_{1} S_{2} S_{1} S_{4} S_{6} S_{6} \\ S_{3} S_{5} S_{5} S_{1} S_{3} S_{6} S_{4} S_{7} S_{7} S_{7} \\ S_{6} S_{4} S_{2} S_{2} S_{5} \\ S_{3} \\ \end{array} \end{array}[/math]

 

[math] \begin{array}{ccc} \begin{array}{c} S_{4} \\ S_{2} S_{7} S_{7} S_{3} S_{6} \\ S_{1} S_{5} S_{4} S_{2} S_{4} S_{3} S_{5} S_{5} S_{6} S_{6} \\ S_{4} S_{3} S_{7} S_{1} S_{7} S_{4} S_{1} S_{3} S_{2} S_{2} \\ S_{6} S_{7} S_{1} S_{6} S_{5} \\ S_{4} \\ \end{array} & \begin{array}{c} S_{5} \\ S_{4} S_{3} S_{2} S_{1} S_{2} \\ S_{5} S_{3} S_{6} S_{5} S_{6} S_{1} S_{1} S_{3} S_{7} S_{7} \\ S_{4} S_{6} S_{2} S_{5} S_{2} S_{7} S_{5} S_{1} S_{4} S_{4} \\ S_{6} S_{3} S_{7} S_{3} S_{2} \\ S_{5} \\ \end{array} & \begin{array}{c} S_{6} \\ S_{3} S_{5} S_{7} S_{4} S_{4} \\ S_{1} S_{6} S_{2} S_{2} S_{3} S_{2} S_{5} S_{6} S_{7} S_{7} \\ S_{3} S_{6} S_{1} S_{5} S_{6} S_{1} S_{5} S_{3} S_{4} S_{4} \\ S_{1} S_{7} S_{2} S_{7} S_{4} \\ S_{6} \\ \end{array} & \begin{array}{c} S_{7} \\ S_{2} S_{4} S_{6} S_{5} S_{6} \\ S_{4} S_{3} S_{7} S_{7} S_{1} S_{5} S_{2} S_{4} S_{3} S_{3} \\ S_{1} S_{7} S_{6} S_{2} S_{1} S_{6} S_{2} S_{7} S_{5} S_{5} \\ S_{4} S_{4} S_{1} S_{6} S_{3} \\ S_{7} \\ \end{array} \end{array} [/math]

Irreducible characters

All irreducible characters have height zero.

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