http://wiki.manchester.ac.uk/blocks/index.php?action=history&feed=atom&title=M%2832%2C51%2C13%29M(32,51,13) - Revision history2026-04-09T00:53:47ZRevision history for this page on the wikiMediaWiki 1.30.1http://wiki.manchester.ac.uk/blocks/index.php?title=M(32,51,13)&diff=1020&oldid=prevCesareGArdito at 15:04, 8 December 20192019-12-08T15:04:43Z<p></p>
<table class="diff diff-contentalign-left" data-mw="interface">
<col class="diff-marker" />
<col class="diff-content" />
<col class="diff-marker" />
<col class="diff-content" />
<tr style="vertical-align: top;" lang="en-GB">
<td colspan="2" style="background-color: white; color:black; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: white; color:black; text-align: center;">Revision as of 15:04, 8 December 2019</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l37" >Line 37:</td>
<td colspan="2" class="diff-lineno">Line 37:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>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>.</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>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>.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>If <math>b</math> is in M(32,51,13), then <math>B</math> is in [[M(32,51,2)]], [[M(32,51,6)]], M(32,51,13) <del class="diffchange diffchange-inline">or </del>[[M(32,51,24)]].</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>If <math>b</math> is in M(32,51,13), then <math>B</math> is in [[M(32,51,2)]], [[M(32,51,6)]], M(32,51,13)<ins class="diffchange diffchange-inline">, </ins>[[M(32,51,24<ins class="diffchange diffchange-inline">)]] or [[M(31,51,33</ins>)]].</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>== Projective indecomposable modules ==</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>== Projective indecomposable modules ==</div></td></tr>
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</table>CesareGArditohttp://wiki.manchester.ac.uk/blocks/index.php?title=M(32,51,13)&diff=1017&oldid=prevCesareGArdito at 14:57, 8 December 20192019-12-08T14:57:27Z<p></p>
<table class="diff diff-contentalign-left" data-mw="interface">
<col class="diff-marker" />
<col class="diff-content" />
<col class="diff-marker" />
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<tr style="vertical-align: top;" lang="en-GB">
<td colspan="2" style="background-color: white; color:black; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: white; color:black; text-align: center;">Revision as of 14:57, 8 December 2019</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l41" >Line 41:</td>
<td colspan="2" class="diff-lineno">Line 41:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>== Projective indecomposable modules ==</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>== Projective indecomposable modules ==</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Labelling the simple <math>B</math>-modules by <math>S_1, \dots, <del class="diffchange diffchange-inline">S_21</del></math>, the projective indecomposable modules have Loewy structure as follows:</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Labelling the simple <math>B</math>-modules by <math>S_1, \dots, <ins class="diffchange diffchange-inline">S_{21}</ins></math>, the projective indecomposable modules have Loewy structure as follows:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div><math>\begin{array}{cccc}</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div><math>\begin{array}{cccc}</div></td></tr>
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</table>CesareGArditohttp://wiki.manchester.ac.uk/blocks/index.php?title=M(32,51,13)&diff=1014&oldid=prevCesareGArdito: /* Covering blocks and covered blocks */2019-12-08T14:50:46Z<p><span dir="auto"><span class="autocomment">Covering blocks and covered blocks</span></span></p>
<table class="diff diff-contentalign-left" data-mw="interface">
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<col class="diff-content" />
<tr style="vertical-align: top;" lang="en-GB">
<td colspan="2" style="background-color: white; color:black; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: white; color:black; text-align: center;">Revision as of 14:50, 8 December 2019</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l37" >Line 37:</td>
<td colspan="2" class="diff-lineno">Line 37:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>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>.</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>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>.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>If <math>b</math> is in M(32,51,<del class="diffchange diffchange-inline">6</del>), then <math>B</math> is in [[M(32,51,<del class="diffchange diffchange-inline">1</del>)]]<del class="diffchange diffchange-inline">, M(32,51,6)</del>, [[M(32,51,<del class="diffchange diffchange-inline">13</del>)]], <del class="diffchange diffchange-inline">[[</del>M(<del class="diffchange diffchange-inline">31</del>,51,<del class="diffchange diffchange-inline">17</del>)<del class="diffchange diffchange-inline">]] </del>or [[M(32,51,<del class="diffchange diffchange-inline">20</del>)]].</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>If <math>b</math> is in M(32,51,<ins class="diffchange diffchange-inline">13</ins>), then <math>B</math> is in [[M(32,51,<ins class="diffchange diffchange-inline">2</ins>)]], [[M(32,51,<ins class="diffchange diffchange-inline">6</ins>)]], M(<ins class="diffchange diffchange-inline">32</ins>,51,<ins class="diffchange diffchange-inline">13</ins>) or [[M(32,51,<ins class="diffchange diffchange-inline">24</ins>)]].</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>== Projective indecomposable modules ==</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>== Projective indecomposable modules ==</div></td></tr>
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</table>CesareGArditohttp://wiki.manchester.ac.uk/blocks/index.php?title=M(32,51,13)&diff=1013&oldid=prevCesareGArdito: Created page with "{{blockbox |title = M(32,51,13) - <math>k(((C_2)^3 : C_7) \times A_4)</math> |image = |representative = <math>k(((C_2)^3 : C_7) \times A_4)</math> |defect = (C2)%5..."2019-12-08T14:48:19Z<p>Created page with "{{blockbox |title = M(32,51,13) - <math>k(((C_2)^3 : C_7) \times A_4)</math> |image = |representative = <math>k(((C_2)^3 : C_7) \times A_4)</math> |defect = (C2)%5..."</p>
<p><b>New page</b></p><div>{{blockbox<br />
|title = M(32,51,13) - <math>k(((C_2)^3 : C_7) \times A_4)</math> <br />
|image = &nbsp; <br />
|representative = <math>k(((C_2)^3 : C_7) \times A_4)</math><br />
|defect = [[(C2)%5E5|<math>(C_2)^5</math>]]<br />
|inertialquotients = <math>C_{21}</math><br />
|k(B) = 32<br />
|l(B) = 21<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)^3 : C_7) \times A_4)</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(32,51,14)]], [[M(32,51,15)]], [[M(32,51,16)]]<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(32,51,6), 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)]].<br />
<br />
== Projective indecomposable modules ==<br />
<br />
Labelling the simple <math>B</math>-modules by <math>S_1, \dots, S_21</math>, the projective indecomposable modules have Loewy structure as follows:<br />
<br />
<math>\begin{array}{cccc}<br />
\begin{array}{c}<br />
S_{1} \\<br />
S_{20} S_{3} S_{19} S_{2} S_{21} \\<br />
S_{14} S_{12} S_{4} S_{5} S_{16} S_{13} S_{18} S_{15} S_{17} S_{1} \\<br />
S_{10} S_{3} S_{20} S_{7} S_{9} S_{19} S_{6} S_{1} S_{8} S_{11} \\<br />
S_{2} S_{21} S_{13} S_{12} S_{5} \\<br />
S_{1} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{2} \\<br />
S_{15} S_{4} S_{17} S_{1} S_{21} \\<br />
S_{18} S_{8} S_{3} S_{6} S_{20} S_{14} S_{16} S_{19} S_{10} S_{2} \\<br />
S_{7} S_{15} S_{17} S_{11} S_{2} S_{4} S_{5} S_{12} S_{13} S_{9} \\<br />
S_{1} S_{21} S_{10} S_{8} S_{6} \\<br />
S_{2} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{3} \\<br />
S_{13} S_{5} S_{19} S_{4} S_{16} \\<br />
S_{3} S_{9} S_{11} S_{20} S_{17} S_{12} S_{1} S_{14} S_{8} S_{6} \\<br />
S_{15} S_{7} S_{13} S_{5} S_{2} S_{3} S_{19} S_{10} S_{18} S_{21} \\<br />
S_{16} S_{4} S_{1} S_{20} S_{12} \\<br />
S_{3} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{4} \\<br />
S_{8} S_{6} S_{17} S_{16} S_{3} \\<br />
S_{2} S_{15} S_{14} S_{10} S_{5} S_{13} S_{19} S_{9} S_{11} S_{4} \\<br />
S_{1} S_{18} S_{20} S_{6} S_{8} S_{17} S_{4} S_{12} S_{7} S_{21} \\<br />
S_{3} S_{16} S_{2} S_{15} S_{10} \\<br />
S_{4} \\<br />
\end{array}<br />
\end{array}</math><br />
<br />
<br>&nbsp;<br><br />
<br />
<math><br />
\begin{array}{cccc}<br />
\begin{array}{c}<br />
S_{5} \\<br />
S_{1} S_{20} S_{13} S_{9} S_{6} \\<br />
S_{5} S_{8} S_{2} S_{18} S_{12} S_{3} S_{19} S_{21} S_{11} S_{15} \\<br />
S_{13} S_{7} S_{4} S_{20} S_{17} S_{5} S_{10} S_{14} S_{16} S_{1} \\<br />
S_{6} S_{9} S_{12} S_{3} S_{19} \\<br />
S_{5} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{6} \\<br />
S_{8} S_{15} S_{2} S_{5} S_{9} \\<br />
S_{4} S_{11} S_{20} S_{21} S_{10} S_{13} S_{1} S_{17} S_{18} S_{6} \\<br />
S_{2} S_{12} S_{7} S_{3} S_{8} S_{6} S_{15} S_{16} S_{19} S_{14} \\<br />
S_{5} S_{9} S_{4} S_{10} S_{17} \\<br />
S_{6} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{7} \\<br />
S_{9} S_{16} S_{21} S_{12} S_{10} \\<br />
S_{5} S_{3} S_{11} S_{1} S_{6} S_{18} S_{4} S_{14} S_{2} S_{7} \\<br />
S_{9} S_{7} S_{20} S_{15} S_{17} S_{16} S_{21} S_{8} S_{19} S_{13} \\<br />
S_{10} S_{12} S_{18} S_{11} S_{14} \\<br />
S_{7} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{8} \\<br />
S_{2} S_{17} S_{10} S_{11} S_{13} \\<br />
S_{12} S_{7} S_{21} S_{14} S_{1} S_{15} S_{6} S_{19} S_{4} S_{8} \\<br />
S_{16} S_{18} S_{10} S_{3} S_{17} S_{8} S_{5} S_{20} S_{9} S_{2} \\<br />
S_{13} S_{11} S_{4} S_{15} S_{6} \\<br />
S_{8} \\<br />
\end{array}<br />
\end{array}<br />
</math><br />
<br />
<br>&nbsp;<br><br />
<br />
<math><br />
\begin{array}{cccc}<br />
\begin{array}{c}<br />
S_{9} \\<br />
S_{21} S_{11} S_{18} S_{5} S_{6} \\<br />
S_{13} S_{14} S_{15} S_{20} S_{8} S_{16} S_{7} S_{2} S_{1} S_{9} \\<br />
S_{12} S_{3} S_{10} S_{4} S_{11} S_{9} S_{19} S_{17} S_{21} S_{18} \\<br />
S_{6} S_{5} S_{7} S_{16} S_{14} \\<br />
S_{9} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{10} \\<br />
S_{6} S_{2} S_{4} S_{12} S_{7} \\<br />
S_{8} S_{5} S_{15} S_{16} S_{1} S_{9} S_{3} S_{21} S_{17} S_{10} \\<br />
S_{10} S_{6} S_{18} S_{2} S_{14} S_{13} S_{20} S_{4} S_{11} S_{19} \\<br />
S_{7} S_{12} S_{17} S_{15} S_{8} \\<br />
S_{10} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{11} \\<br />
S_{7} S_{21} S_{14} S_{8} S_{13} \\<br />
S_{17} S_{12} S_{10} S_{18} S_{2} S_{19} S_{1} S_{9} S_{16} S_{11} \\<br />
S_{11} S_{5} S_{21} S_{7} S_{4} S_{20} S_{14} S_{15} S_{3} S_{6} \\<br />
S_{8} S_{13} S_{18} S_{9} S_{16} \\<br />
S_{11} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{12} \\<br />
S_{5} S_{1} S_{3} S_{10} S_{7} \\<br />
S_{16} S_{20} S_{19} S_{21} S_{2} S_{4} S_{9} S_{13} S_{6} S_{12} \\<br />
S_{14} S_{18} S_{11} S_{15} S_{17} S_{1} S_{8} S_{5} S_{3} S_{12} \\<br />
S_{7} S_{10} S_{19} S_{13} S_{20} \\<br />
S_{12} \\<br />
\end{array}<br />
\end{array}<br />
</math><br />
<br />
<br>&nbsp;<br><br />
<br />
<math><br />
\begin{array}{cccc}<br />
\begin{array}{c}<br />
S_{13} \\<br />
S_{1} S_{12} S_{19} S_{8} S_{11} \\<br />
S_{2} S_{5} S_{21} S_{10} S_{20} S_{7} S_{3} S_{14} S_{17} S_{13} \\<br />
S_{19} S_{16} S_{4} S_{12} S_{15} S_{9} S_{1} S_{18} S_{13} S_{6} \\<br />
S_{8} S_{11} S_{3} S_{20} S_{5} \\<br />
S_{13} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{14} \\<br />
S_{7} S_{9} S_{18} S_{17} S_{19} \\<br />
S_{14} S_{6} S_{5} S_{21} S_{20} S_{11} S_{16} S_{12} S_{15} S_{10} \\<br />
S_{13} S_{3} S_{1} S_{18} S_{7} S_{8} S_{4} S_{14} S_{9} S_{2} \\<br />
S_{17} S_{19} S_{11} S_{21} S_{16} \\<br />
S_{14} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{15} \\<br />
S_{4} S_{10} S_{8} S_{20} S_{18} \\<br />
S_{12} S_{17} S_{13} S_{11} S_{3} S_{7} S_{6} S_{16} S_{2} S_{15} \\<br />
S_{5} S_{4} S_{14} S_{1} S_{9} S_{19} S_{15} S_{10} S_{8} S_{21} \\<br />
S_{20} S_{18} S_{17} S_{6} S_{2} \\<br />
S_{15} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{16} \\<br />
S_{14} S_{11} S_{9} S_{3} S_{4} \\<br />
S_{7} S_{13} S_{21} S_{19} S_{6} S_{17} S_{18} S_{5} S_{8} S_{16} \\<br />
S_{15} S_{16} S_{14} S_{11} S_{9} S_{1} S_{12} S_{10} S_{20} S_{2} \\<br />
S_{3} S_{4} S_{18} S_{21} S_{7} \\<br />
S_{16} \\<br />
\end{array}<br />
\end{array}<br />
</math><br />
<br />
<br>&nbsp;<br><br />
<br />
<math><br />
\begin{array}{ccccc}<br />
\begin{array}{c}<br />
S_{17} \\<br />
S_{15} S_{10} S_{6} S_{19} S_{14} \\<br />
S_{17} S_{18} S_{12} S_{8} S_{5} S_{4} S_{2} S_{20} S_{7} S_{9} \\<br />
S_{3} S_{11} S_{21} S_{15} S_{10} S_{1} S_{17} S_{16} S_{6} S_{13} \\<br />
S_{14} S_{19} S_{2} S_{8} S_{4} \\<br />
S_{17} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{18} \\<br />
S_{16} S_{11} S_{7} S_{20} S_{15} \\<br />
S_{13} S_{14} S_{9} S_{4} S_{8} S_{3} S_{12} S_{10} S_{21} S_{18} \\<br />
S_{7} S_{6} S_{2} S_{18} S_{1} S_{11} S_{5} S_{17} S_{16} S_{19} \\<br />
S_{15} S_{20} S_{9} S_{14} S_{21} \\<br />
S_{18} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{19} \\<br />
S_{12} S_{20} S_{5} S_{14} S_{17} \\<br />
S_{13} S_{7} S_{6} S_{9} S_{3} S_{10} S_{18} S_{1} S_{15} S_{19} \\<br />
S_{8} S_{2} S_{21} S_{4} S_{12} S_{20} S_{19} S_{16} S_{11} S_{5} \\<br />
S_{17} S_{14} S_{3} S_{1} S_{13} \\<br />
S_{19} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{20} \\<br />
S_{12} S_{3} S_{13} S_{18} S_{15} \\<br />
S_{5} S_{8} S_{10} S_{4} S_{11} S_{7} S_{16} S_{1} S_{19} S_{20} \\<br />
S_{2} S_{6} S_{9} S_{14} S_{12} S_{3} S_{13} S_{20} S_{17} S_{21} \\<br />
S_{15} S_{18} S_{19} S_{1} S_{5} \\<br />
S_{20} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{21} \\<br />
S_{18} S_{16} S_{14} S_{1} S_{2} \\<br />
S_{9} S_{7} S_{20} S_{15} S_{11} S_{17} S_{19} S_{4} S_{3} S_{21} \\<br />
S_{6} S_{10} S_{14} S_{21} S_{12} S_{16} S_{5} S_{8} S_{18} S_{13} \\<br />
S_{2} S_{1} S_{9} S_{7} S_{11} \\<br />
S_{21} \\<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}{ccccccccccccccccccccc}<br />
4 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\<br />
2 & 4 & 2 & 1 & 1 & 1 & 1 & 1 & 1 & 2 & 1 & 1 & 2 & 2 & 1 & 1 & 2 & 2 & 1 & 1 & 2 \\<br />
2 & 2 & 4 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 2 & 2 & 1 & 1 & 2 & 2 & 1 & 1 & 2 & 2 & 1 \\<br />
2 & 1 & 1 & 4 & 2 & 2 & 2 & 2 & 2 & 1 & 1 & 1 & 2 & 1 & 1 & 1 & 1 & 1 & 2 & 1 & 1 \\<br />
2 & 1 & 1 & 2 & 4 & 2 & 2 & 2 & 2 & 2 & 1 & 1 & 1 & 1 & 2 & 1 & 1 & 1 & 1 & 1 & 1 \\<br />
2 & 1 & 1 & 2 & 2 & 4 & 2 & 2 & 2 & 1 & 1 & 1 & 1 & 1 & 1 & 2 & 1 & 1 & 1 & 1 & 2 \\<br />
2 & 1 & 1 & 2 & 2 & 2 & 4 & 2 & 2 & 1 & 2 & 1 & 1 & 1 & 1 & 1 & 2 & 1 & 1 & 1 & 1 \\<br />
2 & 1 & 1 & 2 & 2 & 2 & 2 & 4 & 2 & 1 & 1 & 2 & 1 & 1 & 1 & 1 & 1 & 2 & 1 & 1 & 1 \\<br />
2 & 1 & 1 & 2 & 2 & 2 & 2 & 2 & 4 & 1 & 1 & 1 & 1 & 2 & 1 & 1 & 1 & 1 & 1 & 2 & 1 \\<br />
1 & 2 & 1 & 1 & 2 & 1 & 1 & 1 & 1 & 4 & 1 & 1 & 2 & 2 & 2 & 1 & 2 & 2 & 1 & 1 & 2 \\<br />
1 & 1 & 2 & 1 & 1 & 1 & 2 & 1 & 1 & 1 & 4 & 2 & 1 & 1 & 2 & 2 & 2 & 1 & 2 & 2 & 1 \\<br />
1 & 1 & 2 & 1 & 1 & 1 & 1 & 2 & 1 & 1 & 2 & 4 & 1 & 1 & 2 & 2 & 1 & 2 & 2 & 2 & 1 \\<br />
1 & 2 & 1 & 2 & 1 & 1 & 1 & 1 & 1 & 2 & 1 & 1 & 4 & 2 & 1 & 1 & 2 & 2 & 2 & 1 & 2 \\<br />
1 & 2 & 1 & 1 & 1 & 1 & 1 & 1 & 2 & 2 & 1 & 1 & 2 & 4 & 1 & 1 & 2 & 2 & 1 & 2 & 2 \\<br />
1 & 1 & 2 & 1 & 2 & 1 & 1 & 1 & 1 & 2 & 2 & 2 & 1 & 1 & 4 & 2 & 1 & 1 & 2 & 2 & 1 \\<br />
1 & 1 & 2 & 1 & 1 & 2 & 1 & 1 & 1 & 1 & 2 & 2 & 1 & 1 & 2 & 4 & 1 & 1 & 2 & 2 & 2 \\<br />
1 & 2 & 1 & 1 & 1 & 1 & 2 & 1 & 1 & 2 & 2 & 1 & 2 & 2 & 1 & 1 & 4 & 2 & 1 & 1 & 2 \\<br />
1 & 2 & 1 & 1 & 1 & 1 & 1 & 2 & 1 & 2 & 1 & 2 & 2 & 2 & 1 & 1 & 2 & 4 & 1 & 1 & 2 \\<br />
1 & 1 & 2 & 2 & 1 & 1 & 1 & 1 & 1 & 1 & 2 & 2 & 2 & 1 & 2 & 2 & 1 & 1 & 4 & 2 & 1 \\<br />
1 & 1 & 2 & 1 & 1 & 1 & 1 & 1 & 2 & 1 & 2 & 2 & 1 & 2 & 2 & 2 & 1 & 1 & 2 & 4 & 1 \\<br />
1 & 2 & 1 & 1 & 1 & 2 & 1 & 1 & 1 & 2 & 1 & 1 & 2 & 2 & 1 & 2 & 2 & 2 & 1 & 1 & 4<br />
\end{array}\right)</math><br />
<br />
== Decomposition matrix ==<br />
<br />
<math>\left( \begin{array}{ccccccccccccccccccccc}<br />
1 & 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 \\<br />
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 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\<br />
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 & 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 & 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 & 1 \\<br />
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 & 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 & 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 & 1 & 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 \\<br />
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 & 1 & 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 \\<br />
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 & 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 & 1 & 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 \\<br />
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\<br />
1 & 1 & 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 & 1 & 0 & 1 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 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 & 1 & 0 & 0 & 1 & 0 & 1 & 0 \\<br />
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 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 & 1 & 0 & 0 & 0 & 0 & 1 \\<br />
1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 \\<br />
0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 0 & 0 \\<br />
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