M(32,51,13)

M(32,51,13) - [math]k(((C_2)^3 : C_7) \times A_4)[/math]

[[File: |250px]]

Representative:	[math]k(((C_2)^3 : C_7) \times A_4)[/math]
Defect groups:	[math](C_2)^5[/math]
Inertial quotients:	[math]C_{21}[/math]
[math]k(B)=[/math]	32
[math]l(B)=[/math]	21
[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) \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(32,51,14), M(32,51,15), M(32,51,16)
[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(32,51,6), then [math]B[/math] is in M(32,51,1), M(32,51,6), M(32,51,13), M(31,51,17) or M(32,51,20).

Projective indecomposable modules

Labelling the simple [math]B[/math]-modules by [math]S_1, \dots, S_21[/math], the projective indecomposable modules have Loewy structure as follows:

[math]\begin{array}{cccc} \begin{array}{c} S_{1} \\ S_{20} S_{3} S_{19} S_{2} S_{21} \\ S_{14} S_{12} S_{4} S_{5} S_{16} S_{13} S_{18} S_{15} S_{17} S_{1} \\ S_{10} S_{3} S_{20} S_{7} S_{9} S_{19} S_{6} S_{1} S_{8} S_{11} \\ S_{2} S_{21} S_{13} S_{12} S_{5} \\ S_{1} \\ \end{array} & \begin{array}{c} S_{2} \\ S_{15} S_{4} S_{17} S_{1} S_{21} \\ S_{18} S_{8} S_{3} S_{6} S_{20} S_{14} S_{16} S_{19} S_{10} S_{2} \\ S_{7} S_{15} S_{17} S_{11} S_{2} S_{4} S_{5} S_{12} S_{13} S_{9} \\ S_{1} S_{21} S_{10} S_{8} S_{6} \\ S_{2} \\ \end{array} & \begin{array}{c} S_{3} \\ S_{13} S_{5} S_{19} S_{4} S_{16} \\ S_{3} S_{9} S_{11} S_{20} S_{17} S_{12} S_{1} S_{14} S_{8} S_{6} \\ S_{15} S_{7} S_{13} S_{5} S_{2} S_{3} S_{19} S_{10} S_{18} S_{21} \\ S_{16} S_{4} S_{1} S_{20} S_{12} \\ S_{3} \\ \end{array} & \begin{array}{c} S_{4} \\ S_{8} S_{6} S_{17} S_{16} S_{3} \\ S_{2} S_{15} S_{14} S_{10} S_{5} S_{13} S_{19} S_{9} S_{11} S_{4} \\ S_{1} S_{18} S_{20} S_{6} S_{8} S_{17} S_{4} S_{12} S_{7} S_{21} \\ S_{3} S_{16} S_{2} S_{15} S_{10} \\ S_{4} \\ \end{array} \end{array}[/math]

 

[math] \begin{array}{cccc} \begin{array}{c} S_{5} \\ S_{1} S_{20} S_{13} S_{9} S_{6} \\ S_{5} S_{8} S_{2} S_{18} S_{12} S_{3} S_{19} S_{21} S_{11} S_{15} \\ S_{13} S_{7} S_{4} S_{20} S_{17} S_{5} S_{10} S_{14} S_{16} S_{1} \\ S_{6} S_{9} S_{12} S_{3} S_{19} \\ S_{5} \\ \end{array} & \begin{array}{c} S_{6} \\ S_{8} S_{15} S_{2} S_{5} S_{9} \\ S_{4} S_{11} S_{20} S_{21} S_{10} S_{13} S_{1} S_{17} S_{18} S_{6} \\ S_{2} S_{12} S_{7} S_{3} S_{8} S_{6} S_{15} S_{16} S_{19} S_{14} \\ S_{5} S_{9} S_{4} S_{10} S_{17} \\ S_{6} \\ \end{array} & \begin{array}{c} S_{7} \\ S_{9} S_{16} S_{21} S_{12} S_{10} \\ S_{5} S_{3} S_{11} S_{1} S_{6} S_{18} S_{4} S_{14} S_{2} S_{7} \\ S_{9} S_{7} S_{20} S_{15} S_{17} S_{16} S_{21} S_{8} S_{19} S_{13} \\ S_{10} S_{12} S_{18} S_{11} S_{14} \\ S_{7} \\ \end{array} & \begin{array}{c} S_{8} \\ S_{2} S_{17} S_{10} S_{11} S_{13} \\ S_{12} S_{7} S_{21} S_{14} S_{1} S_{15} S_{6} S_{19} S_{4} S_{8} \\ S_{16} S_{18} S_{10} S_{3} S_{17} S_{8} S_{5} S_{20} S_{9} S_{2} \\ S_{13} S_{11} S_{4} S_{15} S_{6} \\ S_{8} \\ \end{array} \end{array} [/math]

 

[math] \begin{array}{cccc} \begin{array}{c} S_{9} \\ S_{21} S_{11} S_{18} S_{5} S_{6} \\ S_{13} S_{14} S_{15} S_{20} S_{8} S_{16} S_{7} S_{2} S_{1} S_{9} \\ S_{12} S_{3} S_{10} S_{4} S_{11} S_{9} S_{19} S_{17} S_{21} S_{18} \\ S_{6} S_{5} S_{7} S_{16} S_{14} \\ S_{9} \\ \end{array} & \begin{array}{c} S_{10} \\ S_{6} S_{2} S_{4} S_{12} S_{7} \\ S_{8} S_{5} S_{15} S_{16} S_{1} S_{9} S_{3} S_{21} S_{17} S_{10} \\ S_{10} S_{6} S_{18} S_{2} S_{14} S_{13} S_{20} S_{4} S_{11} S_{19} \\ S_{7} S_{12} S_{17} S_{15} S_{8} \\ S_{10} \\ \end{array} & \begin{array}{c} S_{11} \\ S_{7} S_{21} S_{14} S_{8} S_{13} \\ S_{17} S_{12} S_{10} S_{18} S_{2} S_{19} S_{1} S_{9} S_{16} S_{11} \\ S_{11} S_{5} S_{21} S_{7} S_{4} S_{20} S_{14} S_{15} S_{3} S_{6} \\ S_{8} S_{13} S_{18} S_{9} S_{16} \\ S_{11} \\ \end{array} & \begin{array}{c} S_{12} \\ S_{5} S_{1} S_{3} S_{10} S_{7} \\ S_{16} S_{20} S_{19} S_{21} S_{2} S_{4} S_{9} S_{13} S_{6} S_{12} \\ S_{14} S_{18} S_{11} S_{15} S_{17} S_{1} S_{8} S_{5} S_{3} S_{12} \\ S_{7} S_{10} S_{19} S_{13} S_{20} \\ S_{12} \\ \end{array} \end{array} [/math]

 

[math] \begin{array}{cccc} \begin{array}{c} S_{13} \\ S_{1} S_{12} S_{19} S_{8} S_{11} \\ S_{2} S_{5} S_{21} S_{10} S_{20} S_{7} S_{3} S_{14} S_{17} S_{13} \\ S_{19} S_{16} S_{4} S_{12} S_{15} S_{9} S_{1} S_{18} S_{13} S_{6} \\ S_{8} S_{11} S_{3} S_{20} S_{5} \\ S_{13} \\ \end{array} & \begin{array}{c} S_{14} \\ S_{7} S_{9} S_{18} S_{17} S_{19} \\ S_{14} S_{6} S_{5} S_{21} S_{20} S_{11} S_{16} S_{12} S_{15} S_{10} \\ S_{13} S_{3} S_{1} S_{18} S_{7} S_{8} S_{4} S_{14} S_{9} S_{2} \\ S_{17} S_{19} S_{11} S_{21} S_{16} \\ S_{14} \\ \end{array} & \begin{array}{c} S_{15} \\ S_{4} S_{10} S_{8} S_{20} S_{18} \\ S_{12} S_{17} S_{13} S_{11} S_{3} S_{7} S_{6} S_{16} S_{2} S_{15} \\ S_{5} S_{4} S_{14} S_{1} S_{9} S_{19} S_{15} S_{10} S_{8} S_{21} \\ S_{20} S_{18} S_{17} S_{6} S_{2} \\ S_{15} \\ \end{array} & \begin{array}{c} S_{16} \\ S_{14} S_{11} S_{9} S_{3} S_{4} \\ S_{7} S_{13} S_{21} S_{19} S_{6} S_{17} S_{18} S_{5} S_{8} S_{16} \\ S_{15} S_{16} S_{14} S_{11} S_{9} S_{1} S_{12} S_{10} S_{20} S_{2} \\ S_{3} S_{4} S_{18} S_{21} S_{7} \\ S_{16} \\ \end{array} \end{array} [/math]

 

[math] \begin{array}{ccccc} \begin{array}{c} S_{17} \\ S_{15} S_{10} S_{6} S_{19} S_{14} \\ S_{17} S_{18} S_{12} S_{8} S_{5} S_{4} S_{2} S_{20} S_{7} S_{9} \\ S_{3} S_{11} S_{21} S_{15} S_{10} S_{1} S_{17} S_{16} S_{6} S_{13} \\ S_{14} S_{19} S_{2} S_{8} S_{4} \\ S_{17} \\ \end{array} & \begin{array}{c} S_{18} \\ S_{16} S_{11} S_{7} S_{20} S_{15} \\ S_{13} S_{14} S_{9} S_{4} S_{8} S_{3} S_{12} S_{10} S_{21} S_{18} \\ S_{7} S_{6} S_{2} S_{18} S_{1} S_{11} S_{5} S_{17} S_{16} S_{19} \\ S_{15} S_{20} S_{9} S_{14} S_{21} \\ S_{18} \\ \end{array} & \begin{array}{c} S_{19} \\ S_{12} S_{20} S_{5} S_{14} S_{17} \\ S_{13} S_{7} S_{6} S_{9} S_{3} S_{10} S_{18} S_{1} S_{15} S_{19} \\ S_{8} S_{2} S_{21} S_{4} S_{12} S_{20} S_{19} S_{16} S_{11} S_{5} \\ S_{17} S_{14} S_{3} S_{1} S_{13} \\ S_{19} \\ \end{array} & \begin{array}{c} S_{20} \\ S_{12} S_{3} S_{13} S_{18} S_{15} \\ S_{5} S_{8} S_{10} S_{4} S_{11} S_{7} S_{16} S_{1} S_{19} S_{20} \\ S_{2} S_{6} S_{9} S_{14} S_{12} S_{3} S_{13} S_{20} S_{17} S_{21} \\ S_{15} S_{18} S_{19} S_{1} S_{5} \\ S_{20} \\ \end{array} & \begin{array}{c} S_{21} \\ S_{18} S_{16} S_{14} S_{1} S_{2} \\ S_{9} S_{7} S_{20} S_{15} S_{11} S_{17} S_{19} S_{4} S_{3} S_{21} \\ S_{6} S_{10} S_{14} S_{21} S_{12} S_{16} S_{5} S_{8} S_{18} S_{13} \\ S_{2} S_{1} S_{9} S_{7} S_{11} \\ S_{21} \\ \end{array} \end{array} [/math]

Irreducible characters

All irreducible characters have height zero.

Cartan matrix

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

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