http://wiki.manchester.ac.uk/blocks/index.php?title=M(32,51,22)&feed=atom&action=historyM(32,51,22) - Revision history2024-03-28T21:19:21ZRevision history for this page on the wikiMediaWiki 1.30.1http://wiki.manchester.ac.uk/blocks/index.php?title=M(32,51,22)&diff=1041&oldid=prevCesareGArdito: Created page with "{{blockbox |title = M(32,51,22) - <math>k((C_2)^4 : C_{31})</math> |image = |representative = <math>k((C_2)^4 : C_{15}) \times C_2)</math> |defect = (C2)%5E5|<math..."2019-12-09T11:59:37Z<p>Created page with "{{blockbox |title = M(32,51,22) - <math>k((C_2)^4 : C_{31})</math> |image = |representative = <math>k((C_2)^4 : C_{15}) \times C_2)</math> |defect = (C2)%5E5|<math..."</p>
<p><b>New page</b></p><div>{{blockbox<br />
|title = M(32,51,22) - <math>k((C_2)^4 : C_{31})</math> <br />
|image = &nbsp; <br />
|representative = <math>k((C_2)^4 : C_{15}) \times C_2)</math><br />
|defect = [[(C2)%5E5|<math>(C_2)^5</math>]]<br />
|inertialquotients = <math>C_{31}</math><br />
|k(B) = 32<br />
|l(B) = 31<br />
|k-morita-frob = 1 <br />
|Pic-k= &nbsp;<br />
|cartan = See below.<br />
|defect-morita-inv? = Yes<br />
|inertial-morita-inv? = Yes<br />
|O-morita? = Yes<br />
|O-morita = <math>\mathcal{O} ((C_2)^5 : C_{31})</math><br />
|decomp = See below.<br />
|O-morita-frob = 1<br />
|Pic-O = <br />
|PIgroup = <br />
|source? = No<br />
|sourcereps =<br />
|k-derived-known? = Yes<br />
|k-derived = [[M(32,51,23)]]<br />
|O-derived-known? = Yes<br />
|coveringblocks =<br />
|coveredblocks =<br />
|pcoveringblocks =<br />
}}<br />
<br />
<br />
== Basic algebra ==<br />
<br />
== Other notatable representatives ==<br />
<br />
== Covering blocks and covered blocks ==<br />
<br />
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>.<br />
<br />
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)]].<br />
<br />
== Projective indecomposable modules ==<br />
<br />
Labelling the simple <math>B</math>-modules by <math>S_1, \dots, S_{31}</math>, the projective indecomposable modules have Loewy structure as follows:<br />
<br />
<math>\begin{array}{ccccc}<br />
\begin{array}{c}<br />
S_{1} \\<br />
S_{8} S_{11} S_{9} S_{7} S_{10} \\<br />
S_{13} S_{17} S_{18} S_{15} S_{5} S_{6} S_{14} S_{19} S_{16} S_{12} \\<br />
S_{24} S_{23} S_{21} S_{4} S_{22} S_{3} S_{27} S_{25} S_{20} S_{26} \\<br />
S_{30} S_{2} S_{29} S_{31} S_{28} \\<br />
S_{1} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{2} \\<br />
S_{8} S_{1} S_{26} S_{17} S_{31} \\<br />
S_{21} S_{25} S_{16} S_{30} S_{9} S_{11} S_{7} S_{14} S_{12} S_{10} \\<br />
S_{20} S_{24} S_{23} S_{13} S_{29} S_{19} S_{15} S_{18} S_{5} S_{6} \\<br />
S_{4} S_{3} S_{27} S_{28} S_{22} \\<br />
S_{2} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{3} \\<br />
S_{2} S_{8} S_{23} S_{14} S_{28} \\<br />
S_{1} S_{16} S_{12} S_{17} S_{9} S_{26} S_{27} S_{20} S_{18} S_{31} \\<br />
S_{10} S_{30} S_{15} S_{7} S_{25} S_{22} S_{13} S_{11} S_{21} S_{24} \\<br />
S_{19} S_{4} S_{6} S_{5} S_{29} \\<br />
S_{3} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{4} \\<br />
S_{21} S_{3} S_{29} S_{17} S_{2} \\<br />
S_{1} S_{23} S_{31} S_{19} S_{25} S_{10} S_{26} S_{8} S_{14} S_{28} \\<br />
S_{6} S_{12} S_{30} S_{9} S_{16} S_{7} S_{27} S_{18} S_{11} S_{20} \\<br />
S_{15} S_{13} S_{5} S_{24} S_{22} \\<br />
S_{4} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{5} \\<br />
S_{3} S_{24} S_{8} S_{12} S_{4} \\<br />
S_{15} S_{2} S_{16} S_{14} S_{29} S_{21} S_{17} S_{28} S_{23} S_{9} \\<br />
S_{25} S_{19} S_{1} S_{31} S_{10} S_{18} S_{13} S_{27} S_{26} S_{20} \\<br />
S_{30} S_{6} S_{11} S_{22} S_{7} \\<br />
S_{5} \\<br />
\end{array}<br />
\end{array}<br />
</math><br />
<br />
<br>&nbsp; <br><br />
<br />
<math>\begin{array}{ccccc}<br />
\begin{array}{c}<br />
S_{6} \\<br />
S_{20} S_{22} S_{14} S_{3} S_{5} \\<br />
S_{4} S_{8} S_{23} S_{2} S_{30} S_{28} S_{26} S_{12} S_{24} S_{18} \\<br />
S_{15} S_{21} S_{16} S_{29} S_{9} S_{1} S_{27} S_{17} S_{31} S_{11} \\<br />
S_{13} S_{19} S_{7} S_{10} S_{25} \\<br />
S_{6} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{7} \\<br />
S_{16} S_{5} S_{6} S_{13} S_{8} \\<br />
S_{9} S_{14} S_{24} S_{4} S_{17} S_{3} S_{22} S_{20} S_{12} S_{25} \\<br />
S_{26} S_{15} S_{30} S_{10} S_{23} S_{18} S_{28} S_{21} S_{2} S_{29} \\<br />
S_{31} S_{19} S_{11} S_{27} S_{1} \\<br />
S_{7} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{8} \\<br />
S_{9} S_{16} S_{14} S_{12} S_{17} \\<br />
S_{25} S_{21} S_{20} S_{10} S_{18} S_{23} S_{13} S_{26} S_{24} S_{15} \\<br />
S_{6} S_{28} S_{19} S_{4} S_{30} S_{11} S_{31} S_{22} S_{27} S_{29} \\<br />
S_{3} S_{1} S_{2} S_{5} S_{7} \\<br />
S_{8} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{9} \\<br />
S_{13} S_{17} S_{10} S_{18} S_{15} \\<br />
S_{27} S_{22} S_{21} S_{6} S_{11} S_{25} S_{19} S_{14} S_{4} S_{26} \\<br />
S_{31} S_{5} S_{20} S_{2} S_{29} S_{30} S_{3} S_{23} S_{7} S_{12} \\<br />
S_{8} S_{24} S_{28} S_{16} S_{1} \\<br />
S_{9} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{10} \\<br />
S_{14} S_{6} S_{18} S_{19} S_{11} \\<br />
S_{23} S_{12} S_{3} S_{27} S_{20} S_{22} S_{7} S_{26} S_{5} S_{15} \\<br />
S_{4} S_{30} S_{16} S_{31} S_{28} S_{24} S_{8} S_{13} S_{2} S_{21} \\<br />
S_{1} S_{29} S_{25} S_{17} S_{9} \\<br />
S_{10} \\<br />
\end{array}<br />
\end{array}<br />
</math><br />
<br />
<br>&nbsp; <br><br />
<br />
<math>\begin{array}{ccccc}<br />
\begin{array}{c}<br />
S_{11} \\<br />
S_{12} S_{15} S_{5} S_{7} S_{19} \\<br />
S_{27} S_{23} S_{13} S_{4} S_{21} S_{24} S_{8} S_{3} S_{6} S_{16} \\<br />
S_{9} S_{20} S_{29} S_{22} S_{25} S_{17} S_{14} S_{28} S_{2} S_{31} \\<br />
S_{26} S_{30} S_{1} S_{10} S_{18} \\<br />
S_{11} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{12} \\<br />
S_{16} S_{15} S_{23} S_{24} S_{21} \\<br />
S_{9} S_{25} S_{19} S_{29} S_{13} S_{20} S_{28} S_{31} S_{4} S_{27} \\<br />
S_{6} S_{30} S_{7} S_{2} S_{17} S_{22} S_{3} S_{10} S_{1} S_{18} \\<br />
S_{11} S_{14} S_{26} S_{5} S_{8} \\<br />
S_{12} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{13} \\<br />
S_{22} S_{25} S_{6} S_{4} S_{17} \\<br />
S_{21} S_{10} S_{29} S_{30} S_{20} S_{3} S_{26} S_{5} S_{2} S_{14} \\<br />
S_{31} S_{8} S_{19} S_{11} S_{23} S_{24} S_{18} S_{28} S_{12} S_{1} \\<br />
S_{15} S_{27} S_{16} S_{7} S_{9} \\<br />
S_{13} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{14} \\<br />
S_{20} S_{26} S_{23} S_{18} S_{12} \\<br />
S_{22} S_{15} S_{24} S_{16} S_{11} S_{31} S_{27} S_{30} S_{21} S_{28} \\<br />
S_{4} S_{19} S_{25} S_{7} S_{13} S_{2} S_{29} S_{1} S_{9} S_{5} \\<br />
S_{6} S_{10} S_{8} S_{3} S_{17} \\<br />
S_{14} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{15} \\<br />
S_{27} S_{21} S_{13} S_{19} S_{4} \\<br />
S_{22} S_{3} S_{29} S_{2} S_{25} S_{17} S_{31} S_{23} S_{6} S_{7} \\<br />
S_{14} S_{1} S_{30} S_{5} S_{28} S_{8} S_{16} S_{20} S_{26} S_{10} \\<br />
S_{12} S_{18} S_{9} S_{11} S_{24} \\<br />
S_{15} \\<br />
\end{array}<br />
\end{array}<br />
</math><br />
<br />
<br />
<br>&nbsp; <br><br />
<br />
<math>\begin{array}{ccccc}<br />
\begin{array}{c}<br />
S_{16} \\<br />
S_{9} S_{24} S_{25} S_{20} S_{13} \\<br />
S_{30} S_{18} S_{4} S_{29} S_{22} S_{28} S_{17} S_{15} S_{6} S_{10} \\<br />
S_{19} S_{11} S_{2} S_{26} S_{27} S_{3} S_{21} S_{14} S_{1} S_{5} \\<br />
S_{7} S_{23} S_{8} S_{12} S_{31} \\<br />
S_{16} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{17} \\<br />
S_{10} S_{14} S_{25} S_{26} S_{21} \\<br />
S_{18} S_{30} S_{6} S_{23} S_{20} S_{12} S_{11} S_{29} S_{19} S_{31} \\<br />
S_{3} S_{7} S_{1} S_{5} S_{27} S_{22} S_{28} S_{16} S_{15} S_{24} \\<br />
S_{2} S_{13} S_{9} S_{8} S_{4} \\<br />
S_{17} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{18} \\<br />
S_{22} S_{26} S_{11} S_{27} S_{15} \\<br />
S_{7} S_{2} S_{13} S_{21} S_{4} S_{5} S_{19} S_{31} S_{30} S_{12} \\<br />
S_{8} S_{29} S_{6} S_{25} S_{16} S_{24} S_{23} S_{17} S_{1} S_{3} \\<br />
S_{10} S_{20} S_{28} S_{14} S_{9} \\<br />
S_{18} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{19} \\<br />
S_{23} S_{3} S_{7} S_{27} S_{6} \\<br />
S_{13} S_{8} S_{20} S_{16} S_{14} S_{22} S_{28} S_{31} S_{5} S_{2} \\<br />
S_{18} S_{4} S_{1} S_{26} S_{17} S_{12} S_{25} S_{9} S_{24} S_{30} \\<br />
S_{10} S_{11} S_{29} S_{15} S_{21} \\<br />
S_{19} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{20} \\<br />
S_{30} S_{18} S_{24} S_{22} S_{28} \\<br />
S_{4} S_{9} S_{2} S_{15} S_{5} S_{26} S_{11} S_{1} S_{27} S_{29} \\<br />
S_{17} S_{8} S_{31} S_{12} S_{13} S_{19} S_{21} S_{10} S_{3} S_{7} \\<br />
S_{25} S_{23} S_{16} S_{14} S_{6} \\<br />
S_{20} \\<br />
\end{array}<br />
\end{array}<br />
</math><br />
<br />
<br />
<br>&nbsp; <br><br />
<br />
<math>\begin{array}{ccccc}<br />
\begin{array}{c}<br />
S_{21} \\<br />
S_{29} S_{19} S_{23} S_{25} S_{31} \\<br />
S_{1} S_{6} S_{20} S_{27} S_{28} S_{30} S_{7} S_{16} S_{10} S_{3} \\<br />
S_{14} S_{18} S_{22} S_{13} S_{8} S_{24} S_{9} S_{11} S_{2} S_{5} \\<br />
S_{26} S_{4} S_{15} S_{17} S_{12} \\<br />
S_{21} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{22} \\<br />
S_{4} S_{5} S_{30} S_{26} S_{2} \\<br />
S_{3} S_{11} S_{24} S_{12} S_{21} S_{1} S_{29} S_{31} S_{8} S_{17} \\<br />
S_{9} S_{10} S_{28} S_{14} S_{15} S_{25} S_{16} S_{23} S_{19} S_{7} \\<br />
S_{27} S_{18} S_{6} S_{20} S_{13} \\<br />
S_{22} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{23} \\<br />
S_{27} S_{16} S_{28} S_{20} S_{31} \\<br />
S_{30} S_{18} S_{2} S_{24} S_{13} S_{1} S_{9} S_{25} S_{7} S_{22} \\<br />
S_{5} S_{11} S_{4} S_{26} S_{29} S_{10} S_{15} S_{8} S_{17} S_{6} \\<br />
S_{3} S_{12} S_{14} S_{19} S_{21} \\<br />
S_{23} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{24} \\<br />
S_{15} S_{28} S_{9} S_{29} S_{4} \\<br />
S_{13} S_{3} S_{2} S_{10} S_{18} S_{19} S_{1} S_{17} S_{21} S_{27} \\<br />
S_{6} S_{31} S_{8} S_{26} S_{22} S_{23} S_{14} S_{25} S_{7} S_{11} \\<br />
S_{5} S_{20} S_{16} S_{12} S_{30} \\<br />
S_{24} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{25} \\<br />
S_{30} S_{10} S_{20} S_{6} S_{29} \\<br />
S_{11} S_{19} S_{24} S_{14} S_{1} S_{28} S_{22} S_{18} S_{5} S_{3} \\<br />
S_{8} S_{2} S_{12} S_{26} S_{15} S_{23} S_{7} S_{9} S_{4} S_{27} \\<br />
S_{31} S_{21} S_{13} S_{17} S_{16} \\<br />
S_{25} \\<br />
\end{array}<br />
\end{array}<br />
</math><br />
<br />
<br />
<br>&nbsp; <br><br />
<br />
<math>\begin{array}{ccccc}<br />
\begin{array}{c}<br />
S_{26} \\<br />
S_{11} S_{12} S_{31} S_{21} S_{30} \\<br />
S_{23} S_{15} S_{25} S_{16} S_{29} S_{7} S_{5} S_{19} S_{1} S_{24} \\<br />
S_{9} S_{8} S_{27} S_{28} S_{3} S_{13} S_{6} S_{4} S_{10} S_{20} \\<br />
S_{22} S_{2} S_{17} S_{18} S_{14} \\<br />
S_{26} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{27} \\<br />
S_{31} S_{22} S_{2} S_{13} S_{7} \\<br />
S_{1} S_{8} S_{25} S_{17} S_{6} S_{4} S_{26} S_{16} S_{30} S_{5} \\<br />
S_{12} S_{3} S_{20} S_{9} S_{14} S_{21} S_{29} S_{24} S_{11} S_{10} \\<br />
S_{28} S_{23} S_{15} S_{18} S_{19} \\<br />
S_{27} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{28} \\<br />
S_{18} S_{2} S_{27} S_{9} S_{1} \\<br />
S_{15} S_{7} S_{31} S_{22} S_{13} S_{10} S_{17} S_{11} S_{26} S_{8} \\<br />
S_{21} S_{4} S_{5} S_{30} S_{12} S_{6} S_{19} S_{14} S_{25} S_{16} \\<br />
S_{24} S_{29} S_{3} S_{20} S_{23} \\<br />
S_{28} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{29} \\<br />
S_{3} S_{19} S_{10} S_{1} S_{28} \\<br />
S_{6} S_{18} S_{7} S_{11} S_{8} S_{2} S_{14} S_{23} S_{9} S_{27} \\<br />
S_{26} S_{17} S_{5} S_{20} S_{16} S_{31} S_{15} S_{13} S_{22} S_{12} \\<br />
S_{25} S_{4} S_{21} S_{24} S_{30} \\<br />
S_{29} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{30} \\<br />
S_{11} S_{1} S_{24} S_{5} S_{29} \\<br />
S_{3} S_{7} S_{15} S_{12} S_{19} S_{9} S_{8} S_{10} S_{28} S_{4} \\<br />
S_{6} S_{23} S_{16} S_{14} S_{2} S_{21} S_{27} S_{17} S_{18} S_{13} \\<br />
S_{26} S_{31} S_{25} S_{22} S_{20} \\<br />
S_{30} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{31} \\<br />
S_{7} S_{30} S_{1} S_{16} S_{25} \\<br />
S_{5} S_{13} S_{20} S_{10} S_{6} S_{8} S_{9} S_{29} S_{24} S_{11} \\<br />
S_{28} S_{22} S_{18} S_{15} S_{4} S_{14} S_{12} S_{19} S_{3} S_{17} \\<br />
S_{27} S_{23} S_{2} S_{26} S_{21} \\<br />
S_{31} \\<br />
\end{array}<br />
\end{array}<br />
</math><br />
<br />
== Irreducible characters ==<br />
<br />
All irreducible characters have height zero.<br />
<br />
== Cartan matrix ==<br />
<math>\left( \begin{array}{ccccccccccccccccccccccccccccccc}<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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<br />
\end{array} \right)</math><br />
<br />
== Decomposition matrix ==<br />
<br />
<math>\left( \begin{array}{ccccccccccccccccccccccccccccccc}<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
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 \\<br />
0 & 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 \\<br />
0 & 0 & 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 \\<br />
0 & 0 & 0 & 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 \\<br />
0 & 0 & 0 & 0 & 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 \\<br />
0 & 0 & 0 & 0 & 0 & 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 \\<br />
0 & 0 & 0 & 0 & 0 & 0 & 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 \\<br />
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<br />
\end{array}\right)</math><br />
<br />
[[(C2)%5E5|Back to <math>(C_2)^5</math>]]</div>CesareGArdito