http://wiki.manchester.ac.uk/blocks/index.php?action=history&feed=atom&title=M%2832%2C51%2C24%29M(32,51,24) - Revision history2026-05-04T05:54:38ZRevision history for this page on the wikiMediaWiki 1.30.1http://wiki.manchester.ac.uk/blocks/index.php?title=M(32,51,24)&diff=1046&oldid=prevCesareGArdito: /* Projective indecomposable modules */2019-12-09T13:03:10Z<p><span dir="auto"><span class="autocomment">Projective indecomposable modules</span></span></p>
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<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 13:03, 9 December 2019</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l198" >Line 198:</td>
<td colspan="2" class="diff-lineno">Line 198:</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>S_{3} S_{5} S_{4} S_{12} S_{10} S_{14} S_{15} \\</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>S_{3} S_{5} S_{4} S_{12} S_{10} S_{14} S_{15} \\</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>S_{15} \\</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>S_{15} \\</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><ins style="font-weight: bold; text-decoration: none;">  \end{array}</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>   \end{array}</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>   \end{array}</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></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></math></div></td></tr>
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</table>CesareGArditohttp://wiki.manchester.ac.uk/blocks/index.php?title=M(32,51,24)&diff=1045&oldid=prevCesareGArdito: /* Projective indecomposable modules */2019-12-09T13:02:56Z<p><span dir="auto"><span class="autocomment">Projective indecomposable modules</span></span></p>
<table class="diff diff-contentalign-left" data-mw="interface">
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<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 13:02, 9 December 2019</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l124" >Line 124:</td>
<td colspan="2" class="diff-lineno">Line 124:</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>S_{3} S_{1} S_{11} \\</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>S_{3} S_{1} S_{11} \\</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>S_{8} \\</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>S_{8} \\</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><del style="font-weight: bold; text-decoration: none;">  \end{array}</del></div></td><td colspan="2"> </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>   \end{array}</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>   \end{array}</div></td></tr>
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</table>CesareGArditohttp://wiki.manchester.ac.uk/blocks/index.php?title=M(32,51,24)&diff=1044&oldid=prevCesareGArdito: Created page with "{{blockbox |title = M(32,51,24) - <math>k(((C_2)^3 : (C_7:C_3)) \times A_4)</math> |image = |representative = <math>k(((C_2)^3 : (C_7:C_3)) \times A_4)</math> |defec..."2019-12-09T13:00:41Z<p>Created page with "{{blockbox |title = M(32,51,24) - <math>k(((C_2)^3 : (C_7:C_3)) \times A_4)</math> |image = |representative = <math>k(((C_2)^3 : (C_7:C_3)) \times A_4)</math> |defec..."</p>
<p><b>New page</b></p><div>{{blockbox<br />
|title = M(32,51,24) - <math>k(((C_2)^3 : (C_7:C_3)) \times A_4)</math> <br />
|image = &nbsp; <br />
|representative = <math>k(((C_2)^3 : (C_7:C_3)) \times A_4)</math><br />
|defect = [[(C2)%5E5|<math>(C_2)^5</math>]]<br />
|inertialquotients = <math>(C_{7}:C_3) \times C_3</math><br />
|k(B) = 32<br />
|l(B) = 15<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:C_3)) \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(32,51,25)]], [[M(32,51,26)]], [[M(32,51,27)]], [[M(32,51,28)]], [[M(32,51,29)]]<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(32,51,24), then <math>B</math> is in [[M(32,51,13)]], [[M(32,51,17)]], [[M(32,51,20)]] or M(32,51,24).<br />
<br />
== Projective indecomposable modules ==<br />
<br />
Labelling the simple <math>B</math>-modules by <math>S_1, \dots, S_{15}</math>, the projective indecomposable modules have Loewy structure as follows:<br />
<br />
<math>\begin{array}{ccc}<br />
\begin{array}{c}<br />
S_{1} \\<br />
S_{3} S_{8} S_{12} \\<br />
S_{1} S_{10} S_{13} S_{15} \\<br />
S_{1} S_{14} S_{12} S_{11} \\<br />
S_{3} S_{8} S_{13} \\<br />
S_{1} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{2} \\<br />
S_{6} S_{4} S_{12} \\<br />
S_{2} S_{10} S_{13} S_{15} \\<br />
S_{2} S_{11} S_{12} S_{14} \\<br />
S_{4} S_{6} S_{13} \\<br />
S_{2} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{3} \\<br />
S_{1} S_{8} S_{15} \\<br />
S_{3} S_{14} S_{10} S_{12} \\<br />
S_{3} S_{11} S_{15} S_{13} \\<br />
S_{8} S_{1} S_{14} \\<br />
S_{3} \\<br />
\end{array}<br />
\end{array}</math><br />
<br />
<br>&nbsp;<br><br />
<br />
<math><br />
\begin{array}{ccc}<br />
\begin{array}{c}<br />
S_{4} \\<br />
S_{2} S_{6} S_{15} \\<br />
S_{4} S_{10} S_{14} S_{12} \\<br />
S_{4} S_{11} S_{15} S_{13} \\<br />
S_{6} S_{2} S_{14} \\<br />
S_{4} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{5} \\<br />
S_{9} S_{7} S_{15} \\<br />
S_{5} S_{12} S_{14} S_{10} \\<br />
S_{5} S_{15} S_{13} S_{11} \\<br />
S_{9} S_{7} S_{14} \\<br />
S_{5} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{6} \\<br />
S_{2} S_{4} S_{10} \\<br />
S_{6} S_{11} S_{15} S_{12} \\<br />
S_{6} S_{14} S_{13} S_{10} \\<br />
S_{4} S_{2} S_{11} \\<br />
S_{6} \\<br />
\end{array}<br />
\end{array}</math><br />
<br />
<br>&nbsp;<br><br />
<br />
<math><br />
\begin{array}{ccc}<br />
\begin{array}{c}<br />
S_{7} \\<br />
S_{5} S_{9} S_{10} \\<br />
S_{7} S_{12} S_{11} S_{15} \\<br />
S_{7} S_{10} S_{14} S_{13} \\<br />
S_{5} S_{9} S_{11} \\<br />
S_{7} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{8} \\<br />
S_{3} S_{1} S_{10} \\<br />
S_{8} S_{12} S_{15} S_{11} \\<br />
S_{8} S_{13} S_{10} S_{14} \\<br />
S_{3} S_{1} S_{11} \\<br />
S_{8} \\<br />
\end{array}<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{9} \\<br />
S_{5} S_{7} S_{12} \\<br />
S_{9} S_{10} S_{15} S_{13} \\<br />
S_{9} S_{14} S_{11} S_{12} \\<br />
S_{7} S_{5} S_{13} \\<br />
S_{9} \\<br />
\end{array}<br />
\end{array}</math><br />
<br />
<br>&nbsp;<br><br />
<br />
<math><br />
\begin{array}{ccc}<br />
\begin{array}{c}<br />
S_{10} \\<br />
S_{12} S_{11} S_{15} S_{11} S_{10} \\<br />
S_{8} S_{6} S_{7} S_{11} S_{15} S_{12} S_{14} S_{14} S_{13} S_{13} S_{10} S_{10} \\<br />
S_{4} S_{3} S_{5} S_{2} S_{9} S_{1} S_{14} S_{12} S_{13} S_{15} S_{11} S_{11} S_{10} S_{10} \\<br />
S_{7} S_{8} S_{6} S_{12} S_{15} S_{11} S_{10} \\<br />
S_{10} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{11} \\<br />
S_{7} S_{8} S_{6} S_{13} S_{11} S_{14} S_{10} \\<br />
S_{2} S_{1} S_{5} S_{9} S_{3} S_{4} S_{12} S_{10} S_{13} S_{15} S_{10} S_{14} S_{11} S_{11} \\<br />
S_{6} S_{8} S_{7} S_{14} S_{13} S_{12} S_{15} S_{12} S_{10} S_{15} S_{11} S_{11} \\<br />
S_{14} S_{11} S_{13} S_{10} S_{10} \\<br />
S_{11} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{12} \\<br />
S_{15} S_{12} S_{13} S_{10} S_{13} \\<br />
S_{1} S_{9} S_{2} S_{14} S_{11} S_{14} S_{10} S_{15} S_{13} S_{11} S_{12} S_{12} \\<br />
S_{5} S_{4} S_{8} S_{7} S_{6} S_{3} S_{12} S_{14} S_{10} S_{11} S_{12} S_{15} S_{13} S_{13} \\<br />
S_{2} S_{1} S_{9} S_{15} S_{12} S_{13} S_{10} \\<br />
S_{12} \\<br />
\end{array}<br />
\end{array}<br />
</math><br />
<br />
<br>&nbsp;<br><br />
<br />
<math><br />
\begin{array}{ccc}<br />
\begin{array}{c}<br />
S_{13} \\<br />
S_{2} S_{9} S_{1} S_{13} S_{11} S_{12} S_{14} \\<br />
S_{3} S_{6} S_{5} S_{7} S_{4} S_{8} S_{12} S_{14} S_{15} S_{10} S_{13} S_{13} S_{11} S_{12} \\<br />
S_{2} S_{9} S_{1} S_{13} S_{13} S_{11} S_{14} S_{12} S_{15} S_{15} S_{10} S_{10} \\<br />
S_{14} S_{13} S_{11} S_{12} S_{12} \\<br />
S_{13} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{14} \\<br />
S_{5} S_{4} S_{3} S_{14} S_{15} S_{13} S_{11} \\<br />
S_{8} S_{6} S_{9} S_{1} S_{2} S_{7} S_{11} S_{13} S_{15} S_{10} S_{15} S_{12} S_{14} S_{14} \\<br />
S_{4} S_{3} S_{5} S_{13} S_{11} S_{15} S_{12} S_{12} S_{10} S_{10} S_{14} S_{14} \\<br />
S_{11} S_{14} S_{13} S_{15} S_{15} \\<br />
S_{14} \\<br />
\end{array}<br />
&<br />
\begin{array}{c}<br />
S_{15} \\<br />
S_{10} S_{15} S_{14} S_{14} S_{12} \\<br />
S_{3} S_{5} S_{4} S_{14} S_{13} S_{13} S_{15} S_{15} S_{10} S_{12} S_{11} S_{11} \\<br />
S_{8} S_{7} S_{2} S_{6} S_{1} S_{9} S_{13} S_{14} S_{10} S_{14} S_{11} S_{12} S_{15} S_{15} \\<br />
S_{3} S_{5} S_{4} S_{12} S_{10} S_{14} S_{15} \\<br />
S_{15} \\<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 & 0 & 2 & 0 & 0 & 0 & 0 & 2 & 0 & 1 & 1 & 2 & 2 & 1 & 1 \\<br />
0 & 4 & 0 & 2 & 0 & 2 & 0 & 0 & 0 & 1 & 1 & 2 & 2 & 1 & 1 \\<br />
2 & 0 & 4 & 0 & 0 & 0 & 0 & 2 & 0 & 1 & 1 & 1 & 1 & 2 & 2 \\<br />
0 & 2 & 0 & 4 & 0 & 2 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 2 & 2 \\<br />
0 & 0 & 0 & 0 & 4 & 0 & 2 & 0 & 2 & 1 & 1 & 1 & 1 & 2 & 2 \\<br />
0 & 2 & 0 & 2 & 0 & 4 & 0 & 0 & 0 & 2 & 2 & 1 & 1 & 1 & 1 \\<br />
0 & 0 & 0 & 0 & 2 & 0 & 4 & 0 & 2 & 2 & 2 & 1 & 1 & 1 & 1 \\<br />
2 & 0 & 2 & 0 & 0 & 0 & 0 & 4 & 0 & 2 & 2 & 1 & 1 & 1 & 1 \\<br />
0 & 0 & 0 & 0 & 2 & 0 & 2 & 0 & 4 & 1 & 1 & 2 & 2 & 1 & 1 \\<br />
1 & 1 & 1 & 1 & 1 & 2 & 2 & 2 & 1 & 8 & 6 & 4 & 3 & 3 & 4 \\<br />
1 & 1 & 1 & 1 & 1 & 2 & 2 & 2 & 1 & 6 & 8 & 3 & 4 & 4 & 3 \\<br />
2 & 2 & 1 & 1 & 1 & 1 & 1 & 1 & 2 & 4 & 3 & 8 & 6 & 3 & 4 \\<br />
2 & 2 & 1 & 1 & 1 & 1 & 1 & 1 & 2 & 3 & 4 & 6 & 8 & 4 & 3 \\<br />
1 & 1 & 2 & 2 & 2 & 1 & 1 & 1 & 1 & 3 & 4 & 3 & 4 & 8 & 6 \\<br />
1 & 1 & 2 & 2 & 2 & 1 & 1 & 1 & 1 & 4 & 3 & 4 & 3 & 6 & 8<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 \\<br />
0 & 0 & 0 & 0 & 1 & 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 \\<br />
0 & 0 & 0 & 0 & 0 & 1 & 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 \\<br />
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 1 & 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 \\<br />
1 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\<br />
0 & 1 & 0 & 1 & 0 & 1 & 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 \\<br />
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 & 1 \\<br />
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 & 1 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\<br />
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 1 & 0 & 0 \\<br />
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 \\<br />
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 1 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 \\<br />
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 \\<br />
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 \\<br />
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 \\<br />
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 1 \\<br />
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 1 & 0 \\<br />
1 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 \\<br />
0 & 1 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 \\<br />
0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 <br />
\end{array}\right)</math><br />
<br />
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