http://wiki.manchester.ac.uk/blocks/index.php?title=M(32,51,5)&feed=atom&action=history
M(32,51,5) - Revision history
2024-03-29T13:27:43Z
Revision history for this page on the wiki
MediaWiki 1.30.1
http://wiki.manchester.ac.uk/blocks/index.php?title=M(32,51,5)&diff=1080&oldid=prev
CesareGArdito at 15:54, 9 December 2019
2019-12-09T15:54:32Z
<p></p>
<table class="diff diff-contentalign-left" data-mw="interface">
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<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 15:54, 9 December 2019</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l34" >Line 34:</td>
<td colspan="2" class="diff-lineno">Line 34:</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>}}</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>}}</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>A block with defect group [[(C2)%5E5|<math>(C_2)^5</math>]] and inertial quotient <math>C_5</math> is Morita <del class="diffchange diffchange-inline">equivalent to M(32,51,5)</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>A block with defect group [[(C2)%5E5|<math>(C_2)^5</math>]] and inertial quotient <math>C_5</math> is <ins class="diffchange diffchange-inline">in this </ins>Morita <ins class="diffchange diffchange-inline">equivalence class</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>It is unknown whether this Morita equivalence class contains blocks with inertial quotient <math>C_7:C_3</math> (with action as in [[M(32,51,20)]]).</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>It is unknown whether this Morita equivalence class contains blocks with inertial quotient <math>C_7:C_3</math> (with action as in [[M(32,51,20)]]).</div></td></tr>
</table>
CesareGArdito
http://wiki.manchester.ac.uk/blocks/index.php?title=M(32,51,5)&diff=1079&oldid=prev
CesareGArdito at 15:53, 9 December 2019
2019-12-09T15:53:43Z
<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 15:53, 9 December 2019</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l4" >Line 4:</td>
<td colspan="2" class="diff-lineno">Line 4:</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>|representative =  <math>k((C_2)^4 : C_5) \times C_2)</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>|representative =  <math>k((C_2)^4 : C_5) \times C_2)</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;"><div>|defect = [[(C2)%5E5|<math>(C_2)^5</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>|defect = [[(C2)%5E5|<math>(C_2)^5</math>]]</div></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>|inertialquotients = <math>C_5</math></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>|inertialquotients = <math>C_5</math><ins class="diffchange diffchange-inline">, ?</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;"><div>|k(B) = 16</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>|k(B) = 16</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;"><div>|l(B) = 5</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>|l(B) = 5</div></td></tr>
</table>
CesareGArdito
http://wiki.manchester.ac.uk/blocks/index.php?title=M(32,51,5)&diff=1066&oldid=prev
CesareGArdito at 15:05, 9 December 2019
2019-12-09T15:05:25Z
<p></p>
<table class="diff diff-contentalign-left" data-mw="interface">
<col class="diff-marker" />
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<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 15:05, 9 December 2019</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l34" >Line 34:</td>
<td colspan="2" class="diff-lineno">Line 34:</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>}}</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>}}</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>A block with defect group [[(C2)%5E5|<math>(C_2)^5</math>]] and inertial quotient <math>C_5</math> is Morita equivalent to <del class="diffchange diffchange-inline">[[</del>M(32,51,5)<del class="diffchange diffchange-inline">]]</del>. It is unknown whether this Morita equivalence class contains blocks with inertial quotient <math>C_7:C_3</math> (with action as in [[M(32,51,20)]]).</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>A block with defect group [[(C2)%5E5|<math>(C_2)^5</math>]] and inertial quotient <math>C_5</math> is Morita equivalent to M(32,51,5).  </div></td></tr>
<tr><td colspan="2"> </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> </div></td></tr>
<tr><td colspan="2"> </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>It is unknown whether this Morita equivalence class contains blocks with inertial quotient <math>C_7:C_3</math> (with action as in [[M(32,51,20)]]).</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>== Basic algebra ==</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>== Basic algebra ==</div></td></tr>
</table>
CesareGArdito
http://wiki.manchester.ac.uk/blocks/index.php?title=M(32,51,5)&diff=1064&oldid=prev
CesareGArdito at 14:57, 9 December 2019
2019-12-09T14:57:49Z
<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 14:57, 9 December 2019</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l34" >Line 34:</td>
<td colspan="2" class="diff-lineno">Line 34:</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>}}</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>}}</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 colspan="2"> </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><ins style="font-weight: bold; text-decoration: none;">A block with defect group [[(C2)%5E5|<math>(C_2)^5</math>]] and inertial quotient <math>C_5</math> is Morita equivalent to [[M(32,51,5)]]. It is unknown whether this Morita equivalence class contains blocks with inertial quotient <math>C_7:C_3</math> (with action as in [[M(32,51,20)]]).</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>== Basic algebra ==</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>== Basic algebra ==</div></td></tr>
</table>
CesareGArdito
http://wiki.manchester.ac.uk/blocks/index.php?title=M(32,51,5)&diff=996&oldid=prev
CesareGArdito at 15:36, 5 December 2019
2019-12-05T15:36:18Z
<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 15:36, 5 December 2019</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l1" >Line 1:</td>
<td colspan="2" class="diff-lineno">Line 1:</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>{{blockbox</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>{{blockbox</div></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>|title = M(32,51,<del class="diffchange diffchange-inline">4</del>) - <math>k((C_2)^4 : C_5) \times C_2)</math>  </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>|title = M(32,51,<ins class="diffchange diffchange-inline">5</ins>) - <math>k((C_2)^4 : C_5) \times C_2)</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;"><div>|image = &nbsp;  </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>|image = &nbsp;  </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;"><div>|representative =  <math>k((C_2)^4 : C_5) \times C_2)</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>|representative =  <math>k((C_2)^4 : C_5) \times C_2)</math></div></td></tr>
</table>
CesareGArdito
http://wiki.manchester.ac.uk/blocks/index.php?title=M(32,51,5)&diff=995&oldid=prev
CesareGArdito: Created page with "{{blockbox |title = M(32,51,4) - <math>k((C_2)^4 : C_5) \times C_2)</math> |image = |representative = <math>k((C_2)^4 : C_5) \times C_2)</math> |defect = (C2)%5E5|..."
2019-12-05T15:36:11Z
<p>Created page with "{{blockbox |title = M(32,51,4) - <math>k((C_2)^4 : C_5) \times C_2)</math> |image = |representative = <math>k((C_2)^4 : C_5) \times C_2)</math> |defect = (C2)%5E5|..."</p>
<p><b>New page</b></p><div>{{blockbox<br />
|title = M(32,51,4) - <math>k((C_2)^4 : C_5) \times C_2)</math> <br />
|image = &nbsp; <br />
|representative = <math>k((C_2)^4 : C_5) \times C_2)</math><br />
|defect = [[(C2)%5E5|<math>(C_2)^5</math>]]<br />
|inertialquotients = <math>C_5</math><br />
|k(B) = 16<br />
|l(B) = 5<br />
|k-morita-frob = 1 <br />
|Pic-k= &nbsp;<br />
|cartan = <math>\left( \begin{array}{ccc}<br />
8 & 6 & 6 & 6 & 6 \\<br />
6 & 8 & 6 & 6 & 6 \\<br />
6 & 6 & 8 & 6 & 6 \\<br />
6 & 6 & 6 & 8 & 6 \\<br />
6 & 6 & 6 & 6 & 8<br />
\end{array} \right)</math><br />
|defect-morita-inv? = Yes<br />
|inertial-morita-inv? = Yes<br />
|O-morita? = Yes<br />
|O-morita = <math>\mathcal{O} ((C_2)^4 : C_5) \times C_2)</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 = Forms a derived equivalence class<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(32,51,5), then <math>B</math> is in [[M(32,51,1)]], M(32,51,5), or [[M(32,51,11)]].<br />
<br />
== Projective indecomposable modules ==<br />
<br />
Labelling the simple <math>B</math>-modules by <math>S_1, S_2, S_3, S_4, S_5</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_1 S_5 S_2 S_4 S_3 \\<br />
S_3 S_5 S_4 S_2 S_1 S_4 S_3 S_1 S_5 S_2 \\<br />
S_2 S_2 S_3 S_5 S_4 S_1 S_3 S_5 S_1 S_4 \\<br />
S_4 S_5 S_2 S_3 S_1 \\<br />
S_1 \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_2 \\<br />
S_4 S_2 S_1 S_5 S_3 \\<br />
S_5 S_1 S_3 S_1 S_4 S_4 S_2 S_2 S_5 S_3 \\<br />
S_2 S_3 S_1 S_5 S_4 S_2 S_3 S_4 S_5 S_1 \\<br />
S_3 S_4 S_1 S_5 S_2 \\<br />
S_2 \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_3 \\<br />
S_5 S_4 S_2 S_3 S_1 \\<br />
S_4 S_2 S_3 S_3 S_4 S_1 S_2 S_1 S_5 S_5 \\<br />
S_4 S_2 S_3 S_2 S_1 S_1 S_4 S_5 S_5 S_3 \\<br />
S_5 S_2 S_4 S_1 S_3 \\<br />
S_3 \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_4 \\<br />
S_5 S_1 S_2 S_3 S_4 \\<br />
S_1 S_3 S_2 S_2 S_3 S_5 S_5 S_1 S_4 S_4 \\<br />
S_5 S_3 S_4 S_4 S_3 S_2 S_5 S_2 S_1 S_1 \\<br />
S_3 S_5 S_2 S_1 S_4 \\<br />
S_4 \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_5 \\<br />
S_2 S_4 S_5 S_1 S_3 \\<br />
S_5 S_3 S_5 S_2 S_4 S_2 S_3 S_4 S_1 S_1 \\<br />
S_2 S_4 S_1 S_5 S_2 S_5 S_1 S_4 S_3 S_3 \\<br />
S_2 S_3 S_1 S_4 S_5 \\<br />
S_5 \\<br />
\end{array}<br />
\end{array}<br />
</math><br />
<br />
== Irreducible characters ==<br />
<br />
All irreducible characters have height zero.<br />
<br />
== Decomposition matrix ==<br />
<br />
<math>\left( \begin{array}{ccc}<br />
1 & 0 & 0 & 0 & 0 \\<br />
1 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 1 & 0 \\<br />
0 & 1 & 0 & 0 & 0 \\<br />
0 & 1 & 0 & 0 & 0 \\<br />
0 & 0 & 1 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 1 \\<br />
0 & 0 & 1 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 1 \\<br />
0 & 0 & 0 & 1 & 0 \\<br />
1 & 1 & 1 & 1 & 1 \\<br />
1 & 1 & 1 & 1 & 1 \\<br />
1 & 1 & 1 & 1 & 1 \\<br />
1 & 1 & 1 & 1 & 1 \\<br />
1 & 1 & 1 & 1 & 1 \\<br />
1 & 1 & 1 & 1 & 1 <br />
\end{array}\right)</math><br />
<br />
[[(C2)%5E5|Back to <math>(C_2)^5</math>]]</div>
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