M(32,51,22)

M(32,51,22) - [math]k((C_2)^4 : C_{31})[/math]

[[File: |250px]]

Representative:	[math]k((C_2)^4 : C_{15}) \times C_2)[/math]
Defect groups:	[math](C_2)^5[/math]
Inertial quotients:	[math]C_{31}[/math]
[math]k(B)=[/math]	32
[math]l(B)=[/math]	31
[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)^5 : C_{31})[/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(32,51,23)
[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(32,51,22), then [math]B[/math] is in M(32,51,1), M(32,51,22) or M(32,51,30).

Projective indecomposable modules

Labelling the simple [math]B[/math]-modules by [math]S_1, \dots, S_{31}[/math], the projective indecomposable modules have Loewy structure as follows:

[math]\begin{array}{ccccc} \begin{array}{c} S_{1} \\ S_{8} S_{11} S_{9} S_{7} S_{10} \\ S_{13} S_{17} S_{18} S_{15} S_{5} S_{6} S_{14} S_{19} S_{16} S_{12} \\ S_{24} S_{23} S_{21} S_{4} S_{22} S_{3} S_{27} S_{25} S_{20} S_{26} \\ S_{30} S_{2} S_{29} S_{31} S_{28} \\ S_{1} \\ \end{array} & \begin{array}{c} S_{2} \\ S_{8} S_{1} S_{26} S_{17} S_{31} \\ S_{21} S_{25} S_{16} S_{30} S_{9} S_{11} S_{7} S_{14} S_{12} S_{10} \\ S_{20} S_{24} S_{23} S_{13} S_{29} S_{19} S_{15} S_{18} S_{5} S_{6} \\ S_{4} S_{3} S_{27} S_{28} S_{22} \\ S_{2} \\ \end{array} & \begin{array}{c} S_{3} \\ S_{2} S_{8} S_{23} S_{14} S_{28} \\ S_{1} S_{16} S_{12} S_{17} S_{9} S_{26} S_{27} S_{20} S_{18} S_{31} \\ S_{10} S_{30} S_{15} S_{7} S_{25} S_{22} S_{13} S_{11} S_{21} S_{24} \\ S_{19} S_{4} S_{6} S_{5} S_{29} \\ S_{3} \\ \end{array} & \begin{array}{c} S_{4} \\ S_{21} S_{3} S_{29} S_{17} S_{2} \\ S_{1} S_{23} S_{31} S_{19} S_{25} S_{10} S_{26} S_{8} S_{14} S_{28} \\ S_{6} S_{12} S_{30} S_{9} S_{16} S_{7} S_{27} S_{18} S_{11} S_{20} \\ S_{15} S_{13} S_{5} S_{24} S_{22} \\ S_{4} \\ \end{array} & \begin{array}{c} S_{5} \\ S_{3} S_{24} S_{8} S_{12} S_{4} \\ S_{15} S_{2} S_{16} S_{14} S_{29} S_{21} S_{17} S_{28} S_{23} S_{9} \\ S_{25} S_{19} S_{1} S_{31} S_{10} S_{18} S_{13} S_{27} S_{26} S_{20} \\ S_{30} S_{6} S_{11} S_{22} S_{7} \\ S_{5} \\ \end{array} \end{array} [/math]

 

[math]\begin{array}{ccccc} \begin{array}{c} S_{6} \\ S_{20} S_{22} S_{14} S_{3} S_{5} \\ S_{4} S_{8} S_{23} S_{2} S_{30} S_{28} S_{26} S_{12} S_{24} S_{18} \\ S_{15} S_{21} S_{16} S_{29} S_{9} S_{1} S_{27} S_{17} S_{31} S_{11} \\ S_{13} S_{19} S_{7} S_{10} S_{25} \\ S_{6} \\ \end{array} & \begin{array}{c} S_{7} \\ S_{16} S_{5} S_{6} S_{13} S_{8} \\ S_{9} S_{14} S_{24} S_{4} S_{17} S_{3} S_{22} S_{20} S_{12} S_{25} \\ S_{26} S_{15} S_{30} S_{10} S_{23} S_{18} S_{28} S_{21} S_{2} S_{29} \\ S_{31} S_{19} S_{11} S_{27} S_{1} \\ S_{7} \\ \end{array} & \begin{array}{c} S_{8} \\ S_{9} S_{16} S_{14} S_{12} S_{17} \\ S_{25} S_{21} S_{20} S_{10} S_{18} S_{23} S_{13} S_{26} S_{24} S_{15} \\ S_{6} S_{28} S_{19} S_{4} S_{30} S_{11} S_{31} S_{22} S_{27} S_{29} \\ S_{3} S_{1} S_{2} S_{5} S_{7} \\ S_{8} \\ \end{array} & \begin{array}{c} S_{9} \\ S_{13} S_{17} S_{10} S_{18} S_{15} \\ S_{27} S_{22} S_{21} S_{6} S_{11} S_{25} S_{19} S_{14} S_{4} S_{26} \\ S_{31} S_{5} S_{20} S_{2} S_{29} S_{30} S_{3} S_{23} S_{7} S_{12} \\ S_{8} S_{24} S_{28} S_{16} S_{1} \\ S_{9} \\ \end{array} & \begin{array}{c} S_{10} \\ S_{14} S_{6} S_{18} S_{19} S_{11} \\ S_{23} S_{12} S_{3} S_{27} S_{20} S_{22} S_{7} S_{26} S_{5} S_{15} \\ S_{4} S_{30} S_{16} S_{31} S_{28} S_{24} S_{8} S_{13} S_{2} S_{21} \\ S_{1} S_{29} S_{25} S_{17} S_{9} \\ S_{10} \\ \end{array} \end{array} [/math]

 

[math]\begin{array}{ccccc} \begin{array}{c} S_{11} \\ S_{12} S_{15} S_{5} S_{7} S_{19} \\ S_{27} S_{23} S_{13} S_{4} S_{21} S_{24} S_{8} S_{3} S_{6} S_{16} \\ S_{9} S_{20} S_{29} S_{22} S_{25} S_{17} S_{14} S_{28} S_{2} S_{31} \\ S_{26} S_{30} S_{1} S_{10} S_{18} \\ S_{11} \\ \end{array} & \begin{array}{c} S_{12} \\ S_{16} S_{15} S_{23} S_{24} S_{21} \\ S_{9} S_{25} S_{19} S_{29} S_{13} S_{20} S_{28} S_{31} S_{4} S_{27} \\ S_{6} S_{30} S_{7} S_{2} S_{17} S_{22} S_{3} S_{10} S_{1} S_{18} \\ S_{11} S_{14} S_{26} S_{5} S_{8} \\ S_{12} \\ \end{array} & \begin{array}{c} S_{13} \\ S_{22} S_{25} S_{6} S_{4} S_{17} \\ S_{21} S_{10} S_{29} S_{30} S_{20} S_{3} S_{26} S_{5} S_{2} S_{14} \\ S_{31} S_{8} S_{19} S_{11} S_{23} S_{24} S_{18} S_{28} S_{12} S_{1} \\ S_{15} S_{27} S_{16} S_{7} S_{9} \\ S_{13} \\ \end{array} & \begin{array}{c} S_{14} \\ S_{20} S_{26} S_{23} S_{18} S_{12} \\ S_{22} S_{15} S_{24} S_{16} S_{11} S_{31} S_{27} S_{30} S_{21} S_{28} \\ S_{4} S_{19} S_{25} S_{7} S_{13} S_{2} S_{29} S_{1} S_{9} S_{5} \\ S_{6} S_{10} S_{8} S_{3} S_{17} \\ S_{14} \\ \end{array} & \begin{array}{c} S_{15} \\ S_{27} S_{21} S_{13} S_{19} S_{4} \\ S_{22} S_{3} S_{29} S_{2} S_{25} S_{17} S_{31} S_{23} S_{6} S_{7} \\ S_{14} S_{1} S_{30} S_{5} S_{28} S_{8} S_{16} S_{20} S_{26} S_{10} \\ S_{12} S_{18} S_{9} S_{11} S_{24} \\ S_{15} \\ \end{array} \end{array} [/math]

 

[math]\begin{array}{ccccc} \begin{array}{c} S_{16} \\ S_{9} S_{24} S_{25} S_{20} S_{13} \\ S_{30} S_{18} S_{4} S_{29} S_{22} S_{28} S_{17} S_{15} S_{6} S_{10} \\ S_{19} S_{11} S_{2} S_{26} S_{27} S_{3} S_{21} S_{14} S_{1} S_{5} \\ S_{7} S_{23} S_{8} S_{12} S_{31} \\ S_{16} \\ \end{array} & \begin{array}{c} S_{17} \\ S_{10} S_{14} S_{25} S_{26} S_{21} \\ S_{18} S_{30} S_{6} S_{23} S_{20} S_{12} S_{11} S_{29} S_{19} S_{31} \\ S_{3} S_{7} S_{1} S_{5} S_{27} S_{22} S_{28} S_{16} S_{15} S_{24} \\ S_{2} S_{13} S_{9} S_{8} S_{4} \\ S_{17} \\ \end{array} & \begin{array}{c} S_{18} \\ S_{22} S_{26} S_{11} S_{27} S_{15} \\ S_{7} S_{2} S_{13} S_{21} S_{4} S_{5} S_{19} S_{31} S_{30} S_{12} \\ S_{8} S_{29} S_{6} S_{25} S_{16} S_{24} S_{23} S_{17} S_{1} S_{3} \\ S_{10} S_{20} S_{28} S_{14} S_{9} \\ S_{18} \\ \end{array} & \begin{array}{c} S_{19} \\ S_{23} S_{3} S_{7} S_{27} S_{6} \\ S_{13} S_{8} S_{20} S_{16} S_{14} S_{22} S_{28} S_{31} S_{5} S_{2} \\ S_{18} S_{4} S_{1} S_{26} S_{17} S_{12} S_{25} S_{9} S_{24} S_{30} \\ S_{10} S_{11} S_{29} S_{15} S_{21} \\ S_{19} \\ \end{array} & \begin{array}{c} S_{20} \\ S_{30} S_{18} S_{24} S_{22} S_{28} \\ S_{4} S_{9} S_{2} S_{15} S_{5} S_{26} S_{11} S_{1} S_{27} S_{29} \\ S_{17} S_{8} S_{31} S_{12} S_{13} S_{19} S_{21} S_{10} S_{3} S_{7} \\ S_{25} S_{23} S_{16} S_{14} S_{6} \\ S_{20} \\ \end{array} \end{array} [/math]

 

[math]\begin{array}{ccccc} \begin{array}{c} S_{21} \\ S_{29} S_{19} S_{23} S_{25} S_{31} \\ S_{1} S_{6} S_{20} S_{27} S_{28} S_{30} S_{7} S_{16} S_{10} S_{3} \\ S_{14} S_{18} S_{22} S_{13} S_{8} S_{24} S_{9} S_{11} S_{2} S_{5} \\ S_{26} S_{4} S_{15} S_{17} S_{12} \\ S_{21} \\ \end{array} & \begin{array}{c} S_{22} \\ S_{4} S_{5} S_{30} S_{26} S_{2} \\ S_{3} S_{11} S_{24} S_{12} S_{21} S_{1} S_{29} S_{31} S_{8} S_{17} \\ S_{9} S_{10} S_{28} S_{14} S_{15} S_{25} S_{16} S_{23} S_{19} S_{7} \\ S_{27} S_{18} S_{6} S_{20} S_{13} \\ S_{22} \\ \end{array} & \begin{array}{c} S_{23} \\ S_{27} S_{16} S_{28} S_{20} S_{31} \\ S_{30} S_{18} S_{2} S_{24} S_{13} S_{1} S_{9} S_{25} S_{7} S_{22} \\ S_{5} S_{11} S_{4} S_{26} S_{29} S_{10} S_{15} S_{8} S_{17} S_{6} \\ S_{3} S_{12} S_{14} S_{19} S_{21} \\ S_{23} \\ \end{array} & \begin{array}{c} S_{24} \\ S_{15} S_{28} S_{9} S_{29} S_{4} \\ S_{13} S_{3} S_{2} S_{10} S_{18} S_{19} S_{1} S_{17} S_{21} S_{27} \\ S_{6} S_{31} S_{8} S_{26} S_{22} S_{23} S_{14} S_{25} S_{7} S_{11} \\ S_{5} S_{20} S_{16} S_{12} S_{30} \\ S_{24} \\ \end{array} & \begin{array}{c} S_{25} \\ S_{30} S_{10} S_{20} S_{6} S_{29} \\ S_{11} S_{19} S_{24} S_{14} S_{1} S_{28} S_{22} S_{18} S_{5} S_{3} \\ S_{8} S_{2} S_{12} S_{26} S_{15} S_{23} S_{7} S_{9} S_{4} S_{27} \\ S_{31} S_{21} S_{13} S_{17} S_{16} \\ S_{25} \\ \end{array} \end{array} [/math]

 

[math]\begin{array}{ccccc} \begin{array}{c} S_{26} \\ S_{11} S_{12} S_{31} S_{21} S_{30} \\ S_{23} S_{15} S_{25} S_{16} S_{29} S_{7} S_{5} S_{19} S_{1} S_{24} \\ S_{9} S_{8} S_{27} S_{28} S_{3} S_{13} S_{6} S_{4} S_{10} S_{20} \\ S_{22} S_{2} S_{17} S_{18} S_{14} \\ S_{26} \\ \end{array} & \begin{array}{c} S_{27} \\ S_{31} S_{22} S_{2} S_{13} S_{7} \\ S_{1} S_{8} S_{25} S_{17} S_{6} S_{4} S_{26} S_{16} S_{30} S_{5} \\ S_{12} S_{3} S_{20} S_{9} S_{14} S_{21} S_{29} S_{24} S_{11} S_{10} \\ S_{28} S_{23} S_{15} S_{18} S_{19} \\ S_{27} \\ \end{array} & \begin{array}{c} S_{28} \\ S_{18} S_{2} S_{27} S_{9} S_{1} \\ S_{15} S_{7} S_{31} S_{22} S_{13} S_{10} S_{17} S_{11} S_{26} S_{8} \\ S_{21} S_{4} S_{5} S_{30} S_{12} S_{6} S_{19} S_{14} S_{25} S_{16} \\ S_{24} S_{29} S_{3} S_{20} S_{23} \\ S_{28} \\ \end{array} & \begin{array}{c} S_{29} \\ S_{3} S_{19} S_{10} S_{1} S_{28} \\ S_{6} S_{18} S_{7} S_{11} S_{8} S_{2} S_{14} S_{23} S_{9} S_{27} \\ S_{26} S_{17} S_{5} S_{20} S_{16} S_{31} S_{15} S_{13} S_{22} S_{12} \\ S_{25} S_{4} S_{21} S_{24} S_{30} \\ S_{29} \\ \end{array} & \begin{array}{c} S_{30} \\ S_{11} S_{1} S_{24} S_{5} S_{29} \\ S_{3} S_{7} S_{15} S_{12} S_{19} S_{9} S_{8} S_{10} S_{28} S_{4} \\ S_{6} S_{23} S_{16} S_{14} S_{2} S_{21} S_{27} S_{17} S_{18} S_{13} \\ S_{26} S_{31} S_{25} S_{22} S_{20} \\ S_{30} \\ \end{array} & \begin{array}{c} S_{31} \\ S_{7} S_{30} S_{1} S_{16} S_{25} \\ S_{5} S_{13} S_{20} S_{10} S_{6} S_{8} S_{9} S_{29} S_{24} S_{11} \\ S_{28} S_{22} S_{18} S_{15} S_{4} S_{14} S_{12} S_{19} S_{3} S_{17} \\ S_{27} S_{23} S_{2} S_{26} S_{21} \\ S_{31} \\ \end{array} \end{array} [/math]

Irreducible characters

All irreducible characters have height zero.

Cartan matrix

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

Decomposition matrix

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

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