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<p id="mathjaxlink" class="pcenter"><a href="chap9.html">[MathJax off]</a></p>
<p><a id="X7DBA3A7E81C71A64" name="X7DBA3A7E81C71A64"></a></p>
<div class="ChapSects"><a href="chap9_mj.html#X7DBA3A7E81C71A64">9 <span class="Heading">Cat<span class="SimpleMath">\(^3\)</span>-groups and Crossed cubes</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap9_mj.html#X7CC52AF4840F478E">9.1 <span class="Heading">Functions for (pre-)cat<span class="SimpleMath">\(^3\)</span>-groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9_mj.html#X7828496F7D72E232">9.1-1 Cat3Group</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9_mj.html#X85A5A6967D942463">9.1-2 Front3DimensionalGroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap9_mj.html#X80E074E37D02B2F6">9.2 <span class="Heading">Enumerating cat<span class="SimpleMath">\(^3\)</span>-groups with a given source</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9_mj.html#X83E4A7367DD13598">9.2-1 AllCat3GroupTriples</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap9_mj.html#X7F8538B580847268">9.3 <span class="Heading">
Definition and constructions for cat<span class="SimpleMath">\(^n\)</span>-groups and their morphisms 
</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9_mj.html#X81C3D39A81B20D76">9.3-1 PreCatnGroup</a></span>
</div></div>
</div>

<h3>9 <span class="Heading">Cat<span class="SimpleMath">\(^3\)</span>-groups and Crossed cubes</span></h3>

<p>The term <em>4d-group</em> refers to a set of equivalent categories of which the ones we are most interested are the categories of <em>crossed cubes</em> and <em>cat<span class="SimpleMath">\(^3\)</span>-groups</em>. A <em>4d-mapping</em> is a function between two 4d-groups which preserves all the structure.</p>

<p>The material in this chapter should be considered very experimental. As yet there are no functions for crossed cubes.</p>

<p><a id="X7CC52AF4840F478E" name="X7CC52AF4840F478E"></a></p>

<h4>9.1 <span class="Heading">Functions for (pre-)cat<span class="SimpleMath">\(^3\)</span>-groups</span></h4>

<p>We shall use the following standard orientation of a cat<span class="SimpleMath">\(^3\)</span>-group <span class="SimpleMath">\(\calE\)</span> on a group <span class="SimpleMath">\(G\)</span>. <span class="SimpleMath">\(\calE\)</span> contains <span class="SimpleMath">\(8\)</span> groups; <span class="SimpleMath">\(12\)</span> cat<span class="SimpleMath">\(^1\)</span>-groups and <span class="SimpleMath">\(6\)</span> cat<span class="SimpleMath">\(^2\)</span>-groups forming the vertices; edges and faces of a cube, as shown in the following diagram.</p>

<p class="center">\[

\vcenter{\xymatrix{
 & H \ar[dddd] <+0.5ex> \ar[dddd] <+0.0ex>
     \ar[rrrr] <+0.5ex> \ar[rrrr] <+0.0ex>
     \ar[dr] <+0.6ex> 
   & & & & N \ar[llll] <+0.6ex> 
             \ar[dddd] <+0.6ex> \ar[dddd] <+0.0ex>
             \ar[dr] <+0.6ex> 
           & \\
 & & G \ar[dddd] <+0.5ex> \ar[dddd] <+0.0ex>
       \ar[rrrr] <+0.5ex> \ar[rrrr] <+0.0ex> 
       \ar[ul] <+0.5ex> \ar[ul] <+0.0ex> 
     & & & &  R \ar[llll] <+0.6ex> 
                \ar[dddd] <+0.5ex> \ar[dddd] <+0.0ex> 
                \ar[ul] <+0.5ex> \ar[ul] <+0.0ex>    \\
 & & & & & & \\
 & & & & & & \\
 & M \ar[uuuu] <+0.6ex> 
     \ar[rrrr] <+0.5ex> \ar[rrrr] <+0.0ex> 
     \ar[dr] <+0.6ex>  
   & & & & L \ar[uuuu] <+0.6ex> 
             \ar[llll] <+0.6ex>
             \ar[dr] <+0.6ex>  
           & \\ 
 & & Q \ar[uuuu] <+0.6ex> 
       \ar[rrrr] <+0.5ex> \ar[rrrr] <+0.0ex> 
       \ar[ul] <+0.5ex> \ar[ul] <+0.0ex>  
     & & & & P \ar[uuuu] <+0.6ex>
               \ar[llll] <+0.6ex>  
               \ar[ul] <+0.5ex> \ar[ul] <+0.0ex>  \\ 
}}

\]</p>

<p>By definition, <span class="SimpleMath">\(\calE\)</span> is generated by three commuting cat<span class="SimpleMath">\(^1\)</span>-groups <span class="SimpleMath">\((G \Rightarrow R), (G \Rightarrow Q)\)</span> and <span class="SimpleMath">\((G \Rightarrow H)\)</span>, but it is more convenient to think of <span class="SimpleMath">\(\calE\)</span> as generated by two cat<span class="SimpleMath">\(^2\)</span>-groups</p>


<ul>
<li><p><em>front</em><span class="SimpleMath">\((\calE)\)</span>, generated by <span class="SimpleMath">\((G \Rightarrow R)\)</span> and <span class="SimpleMath">\((G \Rightarrow Q)\)</span>;</p>

</li>
<li><p><em>left</em><span class="SimpleMath">\((\calE)\)</span>, generated by <span class="SimpleMath">\((G \Rightarrow Q)\)</span> and <span class="SimpleMath">\((G \Rightarrow H)\)</span>.</p>

</li>
</ul>
<p>Because the tail, head and embedding maps all commute, it follows that <em>up</em><span class="SimpleMath">\((\calE)\)</span>, generated by <span class="SimpleMath">\((G \Rightarrow H)\)</span> and <span class="SimpleMath">\((G \Rightarrow R)\)</span>, is a third cat<span class="SimpleMath">\(^2\)</span>-group. The three remaining faces (cat<span class="SimpleMath">\(^2\)</span>-groups) <em>right</em><span class="SimpleMath">\((\calE)\)</span>, <em>down</em><span class="SimpleMath">\((\calE)\)</span> and <em>back</em><span class="SimpleMath">\((\calE)\)</span> are then easily constructed. We shall always use the order [<em>front, left, up, right, down, back</em>] for the six faces.</p>

<p><a id="X7828496F7D72E232" name="X7828496F7D72E232"></a></p>

<h5>9.1-1 Cat3Group</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Cat3Group</code>( <var class="Arg">args</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PreCat3Group</code>( <var class="Arg">args</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsCat3Group</code>( <var class="Arg">C</var> )</td><td class="tdright">( property )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PreCat3GroupByPreCat2Groups</code>( <var class="Arg">L</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>The global functions <code class="code">Cat3Group</code> and <code class="code">PreCat3Group</code> normally take as arguments a pair of cat<span class="SimpleMath">\(^2\)</span>-groups [<em>front, left</em>], or a trio of cat<span class="SimpleMath">\(^1\)</span>-groups [<em>front-up, front-left = left-up, left-left</em>].</p>

<p>In subsection <code class="func">AllCat2GroupsIterator</code> (<a href="chap8_mj.html#X7EFCF9697E845B2C"><span class="RefLink">8.6-4</span></a>) the list of pairs <code class="code">CatnGroupLists(d12).pairs</code> contains the three entries <code class="code">[6,8],[8,11]</code> and <code class="code">[6,11]</code>. It follows that the sixth, eighth and eleventh cat<span class="SimpleMath">\(^1\)</span>-groups for <code class="code">d12</code> generate a cat<span class="SimpleMath">\(^3\)</span>-group.</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">alld12 := AllCat1Groups( d12 );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">C68 := Cat2Group( alld12[6], alld12[8] );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">C811 := Cat2Group( alld12[8], alld12[11] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">C3Ga := Cat3Group( C68, C811 );</span>
cat3-group with generating (pre-)cat1-groups:
1 : [d12 => Group( [ (), (1,6)(2,5)(3,4) ] )]
2 : [d12 => Group( [ (1,4)(2,5)(3,6), (1,3)(4,6) ] )]
3 : [d12 => Group( [ (1,5,3)(2,6,4), (1,4)(2,3)(5,6) ] )]
<span class="GAPprompt">gap></span> <span class="GAPinput">C3Gb := Cat3Group( alld12[6], alld12[8], alld12[11] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">C3Ga = C3Gb;</span>
true

</pre></div>

<p><a id="X85A5A6967D942463" name="X85A5A6967D942463"></a></p>

<h5>9.1-2 Front3DimensionalGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Front3DimensionalGroup</code>( <var class="Arg">C3</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Left3DimensionalGroup</code>( <var class="Arg">C3</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Up3DimensionalGroup</code>( <var class="Arg">C3</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Right3DimensionalGroup</code>( <var class="Arg">C3</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Down3DimensionalGroup</code>( <var class="Arg">C3</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Back3DimensionalGroup</code>( <var class="Arg">C3</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>The six faces of a cat<span class="SimpleMath">\(^3\)</span>-group are stored as these attributes.</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">C116 := Cat2Group( alld12[11], alld12[6] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Up3DimensionalGroup( C3Ga ) = C116;</span>
true

</pre></div>

<p><a id="X80E074E37D02B2F6" name="X80E074E37D02B2F6"></a></p>

<h4>9.2 <span class="Heading">Enumerating cat<span class="SimpleMath">\(^3\)</span>-groups with a given source</span></h4>

<p>Once the list <code class="code">CatnGroupLists(G).pairs</code> has been obtained we may seek all triples <span class="SimpleMath">\([i,j],[j,k]\)</span> and <span class="SimpleMath">\([k,i]\)</span> or <span class="SimpleMath">\([i,k]\)</span> of pairs in this list and then, for each such triple, construct a cat<span class="SimpleMath">\(^3\)</span>-group generated by the <span class="SimpleMath">\(i\)</span>-th, <span class="SimpleMath">\(j\)</span>-th and <span class="SimpleMath">\(k\)</span>-th cat<span class="SimpleMath">\(^1\)</span>-group on <span class="SimpleMath">\(G\)</span>.</p>

<p><a id="X83E4A7367DD13598" name="X83E4A7367DD13598"></a></p>

<h5>9.2-1 AllCat3GroupTriples</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AllCat3GroupTriples</code>( <var class="Arg">G</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AllCat3GroupsNumber</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AllCat3Groups</code>( <var class="Arg">G</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>The list of triples returned by the operation <code class="code">AllCat3GroupTriples</code> is saved as <code class="code">CatnGroupLists(G).cat3triples</code>. The length of this list is the number of cat<span class="SimpleMath">\(^3\)</span>-groups on <span class="SimpleMath">\(G\)</span>, and is saved as <code class="code">CatnGroupNumbers(G).cat3</code>.</p>

<p>As yet there is no operation <code class="code">AllCat3GroupsUpToIsomorphism(G)</code>.</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">triples := AllCat3GroupTriples( d12 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">CatnGroupNumbers( d12 ).cat3; </span>
94
<span class="GAPprompt">gap></span> <span class="GAPinput">triples[46];</span>
[ 5, 7, 11 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">alld12 := AllCat1Groups( d12 );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Cat3Group( alld12[5], alld12[7], alld12[11] );</span>
cat3-group with generating (pre-)cat1-groups:
1 : [d12 => Group( [ (), (1,4)(2,3)(5,6) ] )]
2 : [d12 => Group( [ (1,4)(2,5)(3,6), (2,6)(3,5) ] )]
3 : [d12 => Group( [ (1,5,3)(2,6,4), (1,4)(2,3)(5,6) ] )]

</pre></div>

<p><a id="X7F8538B580847268" name="X7F8538B580847268"></a></p>

<h4>9.3 <span class="Heading">
Definition and constructions for cat<span class="SimpleMath">\(^n\)</span>-groups and their morphisms 
</span></h4>

<p>In this chapter and the previous one we are interested in cat<span class="SimpleMath">\(^2\)</span>-groups and cat<span class="SimpleMath">\(^3\)</span>-groups, and it is convenient in this section to give the more general definition. There are three equivalent descriptions of a cat<span class="SimpleMath">\(^n\)</span>-group.</p>

<p>A <em>cat<span class="SimpleMath">\(^n\)</span>-group</em> consists of the following.</p>


<ul>
<li><p><span class="SimpleMath">\(2^n\)</span> groups <span class="SimpleMath">\(G_A\)</span>, one for each subset <span class="SimpleMath">\(A\)</span> of <span class="SimpleMath">\([n]\)</span>, the <em>vertices</em> of an <span class="SimpleMath">\(n\)</span>-cube.</p>

</li>
<li><p>Group homomorphisms forming <span class="SimpleMath">\(n2^{n-1}\)</span> commuting cat<span class="SimpleMath">\(^1\)</span>-groups,</p>

<p class="center">\[
\calC_{A,i} ~=~ (e_{A,i};\; t_{A,i},\; h_{A,i} \ :\  
G_A \to G_{A \setminus \{i\}}), 
\quad\mbox{for all} \quad A \subseteq [n],~ i \in A,  
\]</p>

<p>the <em>edges</em> of the cube.</p>

</li>
<li><p>These cat<span class="SimpleMath">\(^1\)</span>-groups combine (in sets of <span class="SimpleMath">\(4\)</span>) to form <span class="SimpleMath">\(n(n-1)2^{n-3}\)</span> cat<span class="SimpleMath">\(^2\)</span>-groups <span class="SimpleMath">\(\calC_{A,\{i,j\}}\)</span> for all <span class="SimpleMath">\(\{i,j\} \subseteq A \subseteq [n],~ i \neq j\)</span>, the <em>faces</em> of the cube.</p>

</li>
</ul>
<p>Note that, since the <span class="SimpleMath">\(t_{A,i}, h_{A,i}\)</span> and <span class="SimpleMath">\(e_{A,i}\)</span> commute, composite homomorphisms <span class="SimpleMath">\(t_{A,B}, h_{A,B} : G_A \to G_{A \setminus B}\)</span> and <span class="SimpleMath">\(e_{A,B} : G_{A \setminus B} \to G_A\)</span> are well defined for all <span class="SimpleMath">\(B \subseteq A \subseteq [n]\)</span>.</p>

<p>Secondly, we give the simplest of the three descriptions, again adapted from Ellis-Steiner <a href="chapBib_mj.html#biBell:st">[ES87]</a>.</p>

<p>A cat<span class="SimpleMath">\(^n\)</span>-group <span class="SimpleMath">\(\calC\)</span> consists of <span class="SimpleMath">\(2^n\)</span> groups <span class="SimpleMath">\(G_A\)</span>, one for each subset <span class="SimpleMath">\(A\)</span> of <span class="SimpleMath">\([n]\)</span>, and <span class="SimpleMath">\(3n\)</span> homomorphisms</p>

<p class="center">\[
t_{[n],i}, h_{[n],i} : G_{[n]} \to G_{[n] \setminus \{i\}},~ 
 e_{[n],i} : G_{[n] \setminus \{i\}} \to G_{[n]}, 
\]</p>

<p>satisfying the following axioms for all <span class="SimpleMath">\(1 \leqslant i \leqslant n\)</span>,}</p>


<ul>
<li><p>the <span class="SimpleMath">\(\calC_{[n],i} ~=~ (e_{[n],i};\; t_{[n],i},\; h_{[n],i} \ :\ G_{[n]} \to G_{[n] \setminus \{i\}})~\)</span> are <em>commuting</em> cat<span class="SimpleMath">\(^1\)</span>-groups, so that:</p>

</li>
<li><p><span class="SimpleMath">\( (e_1 \circ t_1) \circ (e_2 \circ t_2) = (e_2 \circ t_2) \circ (e_1 \circ t_1), \quad (e_1 \circ h_1) \circ (e_2 \circ h_2) = (e_2 \circ h_2) \circ (e_1 \circ h_1), \)</span></p>

</li>
<li><p><span class="SimpleMath">\( (e_1 \circ t_1) \circ (e_2 \circ h_2) = (e_2 \circ h_2) \circ (e_1 \circ t_1), \quad (e_2 \circ t_2) \circ (e_1 \circ h_1) = (e_1 \circ h_1) \circ (e_2 \circ t_2). \)</span></p>

</li>
</ul>
<p>Our third description defines a cat<span class="SimpleMath">\(^n\)</span>-group as a "cat\(^1\)-group of cat\(^{(n-1)}\)-groups".</p>

<p>A <em>cat<span class="SimpleMath">\(^n\)</span>-group</em> <span class="SimpleMath">\(\calC\)</span> consists of two cat<span class="SimpleMath">\(^{(n-1)}\)</span>-groups:</p>


<ul>
<li><p><span class="SimpleMath">\(\calA\)</span> with groups <span class="SimpleMath">\(G_A,\; A \subseteq [n-1]\)</span>, and homomorphisms <span class="SimpleMath">\(\ddot{t}_{A,i}, \ddot{h}_{A,i}, \ddot{e}_{A,i}\)</span>,</p>

</li>
<li><p><span class="SimpleMath">\(\calB\)</span> with groups <span class="SimpleMath">\(H_B,\; B \subseteq [n-1]\)</span>, and homomorphisms <span class="SimpleMath">\(\dot{t}_{B,i}, \dot{h}_{B,i}, \dot{e}_{B,i}\)</span>, and</p>

</li>
<li><p>cat<span class="SimpleMath">\(^{(n-1)}\)</span>-morphisms <span class="SimpleMath">\(t,h : \calA \to \calB\)</span> and <span class="SimpleMath">\(e : \calB \to \calA\)</span> subject to the following conditions:</p>

<p class="center">\[
(t \circ e) ~\mbox{and}~ (h \circ e)  
~\mbox{are the identity mapping on}~ \calB, \qquad
[\ker t, \ker h] = \{ 1_{\calA} \}.
\]</p>

</li>
</ul>
<p><a id="X81C3D39A81B20D76" name="X81C3D39A81B20D76"></a></p>

<h5>9.3-1 PreCatnGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PreCatnGroup</code>( <var class="Arg">L</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CatnGroup</code>( <var class="Arg">L</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>The operation <code class="code">(Pre)CatnGroup</code> expects as input a list of cat<span class="SimpleMath">\(^1\)</span>-groups. For our group <code class="code">d12</code> we may construct various cat<span class="SimpleMath">\(^4\)</span>-groups, and here is one of them.</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">PC4 := PreCatnGroup( [ alld12[5], alld12[7], alld12[11], alld12[12] ] );</span>
(pre-)cat4-group with generating (pre-)cat1-groups:
1 : [d12 => Group( [ (), (1,4)(2,3)(5,6) ] )]
2 : [d12 => Group( [ (1,4)(2,5)(3,6), (2,6)(3,5) ] )]
3 : [d12 => Group( [ (1,5,3)(2,6,4), (1,4)(2,3)(5,6) ] )]
4 : [d12 => Group( [ (1,2,3,4,5,6), (2,6)(3,5) ] )]
<span class="GAPprompt">gap></span> <span class="GAPinput">IsCatnGroup( PC4 );                                             </span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">HigherDimension( PC4 );</span>
5

</pre></div>

<p>For a cat<span class="SimpleMath">\(^5\)</span>-group we may start with the cyclic group whose order is the product of the first five primes. With this group we may form <span class="SimpleMath">\(32\)</span> cat<span class="SimpleMath">\(^1\)</span>-groups and <span class="SimpleMath">\(528\)</span> cat<span class="SimpleMath">\(^2\)</span>-groups.</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">G := Group( (1,2), (3,4,5), (6,7,8,9,10), (11,12,13,14,15,16,17),</span>
<span class="GAPprompt">></span> <span class="GAPinput">               (20,21,22,23,24,25,26,27,28,29,30) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SetName( G, "C2310" );</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">all1 := AllCat1Groups( G );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Print( "G has ", CatnGroupNumbers( G ).cat1, " cat1-groups\n" );</span>
G has 32 cat1-groups
<span class="GAPprompt">gap></span> <span class="GAPinput">PC5 := PreCatnGroup( [ all1[2], all1[5], all1[13], all1[25], all1[32] ] );</span>
(pre-)cat5-group with generating (pre-)cat1-groups:
1 : [C2310 => Group( [ (), (), (), (), (1,2) ] )]
2 : [C2310 => Group( [ (), (), (), (3,4,5), (1,2) ] )]
3 : [C2310 => Group( [ (), (), ( 6, 7, 8, 9,10), (3,4,5), (1,2) ] )]
4 : [C2310 => Group( [ (), (11,12,13,14,15,16,17), ( 6, 7, 8, 9,10), (3,4,5), 
  (1,2) ] )]
5 : [C2310 => Group( [ (20,21,22,23,24,25,26,27,28,29,30), 
  (11,12,13,14,15,16,17), ( 6, 7, 8, 9,10), (3,4,5), (1,2) ] )]
<span class="GAPprompt">gap></span> <span class="GAPinput">HigherDimension( PC5 );</span>
6

</pre></div>


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