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2019-12-05T15:32:05Z

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{{blockbox
|title = M(32,51,4) - <math>k((C_2)^4 : C_3) \times C_2)</math>
|image =  
|representative = <math>k((C_2)^4 : C_3) \times C_2)</math>
|defect = [[(C2)%5E5|<math>(C_2)^5</math>]]
|inertialquotients = <math>C_3</math>
|k(B) = 16
|l(B) = 3
|k-morita-frob = 1
|Pic-k=  
|cartan = <math>\left( \begin{array}{ccc}
12 & 10 & 10\\
10 & 12 & 10\\
10 & 10 & 12
\end{array} \right)</math>
|defect-morita-inv? = Yes
|inertial-morita-inv? = Yes
|O-morita? = Yes
|O-morita = <math>\mathcal{O} ((C_2)^4 : C_3) \times C_2)</math>
|decomp = See below.
|O-morita-frob = 1
|Pic-O =
|PIgroup =
|source? = No
|sourcereps =
|k-derived-known? = Yes
|k-derived = Forms a derived equivalence class
|O-derived-known? = Yes
|coveringblocks =
|coveredblocks =
|pcoveringblocks =
}}

== Basic algebra ==

== Other notatable representatives ==

== 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(32,51,4), then <math>B</math> is in [[M(32,51,1)]], M(32,51,4), or [[M(32,51,8)]].

== Projective indecomposable modules ==

Labelling the simple <math>B</math>-modules by <math>S_1, S_2, S_3</math>, the projective indecomposable modules have Loewy structure as follows:

<math>\begin{array}{ccc}
\begin{array}{c}
S_1 \\
S_1 S_2 S_3 S_3 S_2 \\
S_3 S_2 S_3 S_2 S_1 S_1 S_1 S_3 S_2 S_1 \\
S_3 S_1 S_2 S_1 S_1 S_1 S_3 S_2 S_3 S_2 \\
S_3 S_2 S_3 S_2 S_1 \\
S_1
\end{array}
&
\begin{array}{c}
S_2 \\
S_1 S_3 S_2 S_1 S_3 \\
S_1 S_1 S_2 S_3 S_3 S_1 S_3 S_2 S_2 S_2 \\
S_3 S_2 S_2 S_2 S_1 S_3 S_1 S_2 S_1 S_3 \\
S_3 S_1 S_3 S_1 S_2 \\
S_2 \\
\end{array}
&
\begin{array}{c}
S_3 \\
S_2 S_1 S_2 S_1 S_3 \\
S_1 S_1 S_3 S_3 S_2 S_2 S_2 S_3 S_1 S_3 \\
S_1 S_3 S_2 S_3 S_1 S_2 S_3 S_3 S_1 S_2 \\
S_1 S_2 S_2 S_1 S_3 \\
S_3 \\
\end{array}
\end{array}
</math>

== Irreducible characters ==

All irreducible characters have height zero.

== Decomposition matrix ==

<math>\left( \begin{array}{ccc}
1 & 0 & 0 \\
1 & 0 & 0 \\
0 & 0 & 1 \\
0 & 0 & 1 \\
0 & 1 & 0 \\
0 & 1 & 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
\end{array}\right)</math>

[[(C2)%5E5|Back to <math>(C_2)^5</math>]]

M(32,51,4) - Revision history

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