M(32,51,24)

M(32,51,24) - [math]k(((C_2)^3 : (C_7:C_3)) \times A_4)[/math]

[[File: |250px]]

Representative:	[math]k(((C_2)^3 : (C_7:C_3)) \times A_4)[/math]
Defect groups:	[math](C_2)^5[/math]
Inertial quotients:	[math](C_{7}:C_3) \times C_3[/math]
[math]k(B)=[/math]	32
[math]l(B)=[/math]	15
[math]{\rm mf}_k(B)=[/math]	1
[math]{\rm Pic}_k(B)=[/math]
Cartan matrix:	See below.
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_2)^3 : (C_7:C_3)) \times A_4)[/math]
Decomposition matrices:	See below.
[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:	M(32,51,25), M(32,51,26), M(32,51,27), M(32,51,28), M(32,51,29)
[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:

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(32,51,24), then [math]B[/math] is in M(32,51,13), M(32,51,17), M(32,51,20) or M(32,51,24).

Projective indecomposable modules

Labelling the simple [math]B[/math]-modules by [math]S_1, \dots, S_{15}[/math], the projective indecomposable modules have Loewy structure as follows:

[math]\begin{array}{ccc} \begin{array}{c} S_{1} \\ S_{3} S_{8} S_{12} \\ S_{1} S_{10} S_{13} S_{15} \\ S_{1} S_{14} S_{12} S_{11} \\ S_{3} S_{8} S_{13} \\ S_{1} \\ \end{array} & \begin{array}{c} S_{2} \\ S_{6} S_{4} S_{12} \\ S_{2} S_{10} S_{13} S_{15} \\ S_{2} S_{11} S_{12} S_{14} \\ S_{4} S_{6} S_{13} \\ S_{2} \\ \end{array} & \begin{array}{c} S_{3} \\ S_{1} S_{8} S_{15} \\ S_{3} S_{14} S_{10} S_{12} \\ S_{3} S_{11} S_{15} S_{13} \\ S_{8} S_{1} S_{14} \\ S_{3} \\ \end{array} \end{array}[/math]

 

[math] \begin{array}{ccc} \begin{array}{c} S_{4} \\ S_{2} S_{6} S_{15} \\ S_{4} S_{10} S_{14} S_{12} \\ S_{4} S_{11} S_{15} S_{13} \\ S_{6} S_{2} S_{14} \\ S_{4} \\ \end{array} & \begin{array}{c} S_{5} \\ S_{9} S_{7} S_{15} \\ S_{5} S_{12} S_{14} S_{10} \\ S_{5} S_{15} S_{13} S_{11} \\ S_{9} S_{7} S_{14} \\ S_{5} \\ \end{array} & \begin{array}{c} S_{6} \\ S_{2} S_{4} S_{10} \\ S_{6} S_{11} S_{15} S_{12} \\ S_{6} S_{14} S_{13} S_{10} \\ S_{4} S_{2} S_{11} \\ S_{6} \\ \end{array} \end{array}[/math]

 

[math] \begin{array}{ccc} \begin{array}{c} S_{7} \\ S_{5} S_{9} S_{10} \\ S_{7} S_{12} S_{11} S_{15} \\ S_{7} S_{10} S_{14} S_{13} \\ S_{5} S_{9} S_{11} \\ S_{7} \\ \end{array} & \begin{array}{c} S_{8} \\ S_{3} S_{1} S_{10} \\ S_{8} S_{12} S_{15} S_{11} \\ S_{8} S_{13} S_{10} S_{14} \\ S_{3} S_{1} S_{11} \\ S_{8} \\ \end{array} & \begin{array}{c} S_{9} \\ S_{5} S_{7} S_{12} \\ S_{9} S_{10} S_{15} S_{13} \\ S_{9} S_{14} S_{11} S_{12} \\ S_{7} S_{5} S_{13} \\ S_{9} \\ \end{array} \end{array}[/math]

 

[math] \begin{array}{ccc} \begin{array}{c} S_{10} \\ S_{12} S_{11} S_{15} S_{11} S_{10} \\ S_{8} S_{6} S_{7} S_{11} S_{15} S_{12} S_{14} S_{14} S_{13} S_{13} S_{10} S_{10} \\ S_{4} S_{3} S_{5} S_{2} S_{9} S_{1} S_{14} S_{12} S_{13} S_{15} S_{11} S_{11} S_{10} S_{10} \\ S_{7} S_{8} S_{6} S_{12} S_{15} S_{11} S_{10} \\ S_{10} \\ \end{array} & \begin{array}{c} S_{11} \\ S_{7} S_{8} S_{6} S_{13} S_{11} S_{14} S_{10} \\ S_{2} S_{1} S_{5} S_{9} S_{3} S_{4} S_{12} S_{10} S_{13} S_{15} S_{10} S_{14} S_{11} S_{11} \\ S_{6} S_{8} S_{7} S_{14} S_{13} S_{12} S_{15} S_{12} S_{10} S_{15} S_{11} S_{11} \\ S_{14} S_{11} S_{13} S_{10} S_{10} \\ S_{11} \\ \end{array} & \begin{array}{c} S_{12} \\ S_{15} S_{12} S_{13} S_{10} S_{13} \\ S_{1} S_{9} S_{2} S_{14} S_{11} S_{14} S_{10} S_{15} S_{13} S_{11} S_{12} S_{12} \\ S_{5} S_{4} S_{8} S_{7} S_{6} S_{3} S_{12} S_{14} S_{10} S_{11} S_{12} S_{15} S_{13} S_{13} \\ S_{2} S_{1} S_{9} S_{15} S_{12} S_{13} S_{10} \\ S_{12} \\ \end{array} \end{array} [/math]

 

[math] \begin{array}{ccc} \begin{array}{c} S_{13} \\ S_{2} S_{9} S_{1} S_{13} S_{11} S_{12} S_{14} \\ S_{3} S_{6} S_{5} S_{7} S_{4} S_{8} S_{12} S_{14} S_{15} S_{10} S_{13} S_{13} S_{11} S_{12} \\ S_{2} S_{9} S_{1} S_{13} S_{13} S_{11} S_{14} S_{12} S_{15} S_{15} S_{10} S_{10} \\ S_{14} S_{13} S_{11} S_{12} S_{12} \\ S_{13} \\ \end{array} & \begin{array}{c} S_{14} \\ S_{5} S_{4} S_{3} S_{14} S_{15} S_{13} S_{11} \\ S_{8} S_{6} S_{9} S_{1} S_{2} S_{7} S_{11} S_{13} S_{15} S_{10} S_{15} S_{12} S_{14} S_{14} \\ S_{4} S_{3} S_{5} S_{13} S_{11} S_{15} S_{12} S_{12} S_{10} S_{10} S_{14} S_{14} \\ S_{11} S_{14} S_{13} S_{15} S_{15} \\ S_{14} \\ \end{array} & \begin{array}{c} S_{15} \\ S_{10} S_{15} S_{14} S_{14} S_{12} \\ S_{3} S_{5} S_{4} S_{14} S_{13} S_{13} S_{15} S_{15} S_{10} S_{12} S_{11} S_{11} \\ S_{8} S_{7} S_{2} S_{6} S_{1} S_{9} S_{13} S_{14} S_{10} S_{14} S_{11} S_{12} S_{15} S_{15} \\ S_{3} S_{5} S_{4} S_{12} S_{10} S_{14} S_{15} \\ S_{15} \\ \end{array} \end{array} [/math]

Irreducible characters

All irreducible characters have height zero.

Cartan matrix

[math]\left( \begin{array}{ccccccccccccccccccccc} 4 & 0 & 2 & 0 & 0 & 0 & 0 & 2 & 0 & 1 & 1 & 2 & 2 & 1 & 1 \\ 0 & 4 & 0 & 2 & 0 & 2 & 0 & 0 & 0 & 1 & 1 & 2 & 2 & 1 & 1 \\ 2 & 0 & 4 & 0 & 0 & 0 & 0 & 2 & 0 & 1 & 1 & 1 & 1 & 2 & 2 \\ 0 & 2 & 0 & 4 & 0 & 2 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 2 & 2 \\ 0 & 0 & 0 & 0 & 4 & 0 & 2 & 0 & 2 & 1 & 1 & 1 & 1 & 2 & 2 \\ 0 & 2 & 0 & 2 & 0 & 4 & 0 & 0 & 0 & 2 & 2 & 1 & 1 & 1 & 1 \\ 0 & 0 & 0 & 0 & 2 & 0 & 4 & 0 & 2 & 2 & 2 & 1 & 1 & 1 & 1 \\ 2 & 0 & 2 & 0 & 0 & 0 & 0 & 4 & 0 & 2 & 2 & 1 & 1 & 1 & 1 \\ 0 & 0 & 0 & 0 & 2 & 0 & 2 & 0 & 4 & 1 & 1 & 2 & 2 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 & 2 & 2 & 2 & 1 & 8 & 6 & 4 & 3 & 3 & 4 \\ 1 & 1 & 1 & 1 & 1 & 2 & 2 & 2 & 1 & 6 & 8 & 3 & 4 & 4 & 3 \\ 2 & 2 & 1 & 1 & 1 & 1 & 1 & 1 & 2 & 4 & 3 & 8 & 6 & 3 & 4 \\ 2 & 2 & 1 & 1 & 1 & 1 & 1 & 1 & 2 & 3 & 4 & 6 & 8 & 4 & 3 \\ 1 & 1 & 2 & 2 & 2 & 1 & 1 & 1 & 1 & 3 & 4 & 3 & 4 & 8 & 6 \\ 1 & 1 & 2 & 2 & 2 & 1 & 1 & 1 & 1 & 4 & 3 & 4 & 3 & 6 & 8 \end{array}\right)[/math]

Decomposition matrix

[math]\left( \begin{array}{ccccccccccccccccccccc} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 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 & 0 & 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 1 & 0 \\ 1 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 \\ 0 & 1 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 \\ 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \end{array}\right)[/math]

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