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<p id="mathjaxlink" class="pcenter"><a href="chap3_mj.html">[MathJax on]</a></p>
<p><a id="X7F4E157A827198EA" name="X7F4E157A827198EA"></a></p>
<div class="ChapSects"><a href="chap3.html#X7F4E157A827198EA">3 <span class="Heading">Compatibility</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X81B0CAE97D161B97">3.1 <span class="Heading">The compatibility condition (C)</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X82C839D7794DDBCD">3.2 <span class="Heading">Compatible elements</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X84AA17C28591AC1F">3.2-1 AreCompatibleBallElements</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X8206962582F9C96A">3.2-2 CompatibleBallElement</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7ECC9E9982597E12">3.2-3 <span class="Heading">CompatibilitySet</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X837E15117DB5B1DE">3.2-4 AssembleAutomorphism</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X7967612B81C5074E">3.3 <span class="Heading">Compatible subgroups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X79D1BC54814345D4">3.3-1 MaximalCompatibleSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X8302F28A85F1C4FE">3.3-2 SatisfiesC</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7FF0262182F3B9E8">3.3-3 CompatibleSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7CEBD97183C3399E">3.3-4 ConjugacyClassRepsCompatibleSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X84F140AC7AD8D9EE">3.3-5 ConjugacyClassRepsCompatibleGroupsWithProjection</a></span>
</div></div>
</div>

<h3>3 <span class="Heading">Compatibility</span></h3>

<p><a id="X81B0CAE97D161B97" name="X81B0CAE97D161B97"></a></p>

<h4>3.1 <span class="Heading">The compatibility condition (C)</span></h4>

<p>A subgroup <span class="Math">F\le\mathrm{Aut}(B_{d,k})</span> satifies the compatibility condition (C) if and only if <span class="Math">\mathrm{U}_{k}(F)</span> is locally action isomorphic to <span class="Math">F</span>, see <a href="chapBib.html#biBTor20">[Tor20, Proposition 3.8]</a>. The term <em>compatibility</em> comes from the following translation of this condition into properties of the <span class="Math">(k-1)</span>-local actions of elements of <span class="Math">F</span>: The group <span class="Math">F</span> satisfies (C) if and only if</p>

<p class="pcenter">\forall \alpha\in F\ \forall\omega\in\Omega\ \exists\beta\in F:\ \sigma_{k-1}(\alpha,b)=\sigma_{k-1}(\beta,b_{\omega}),\ \sigma_{k-1}(\alpha,b_{\omega})=\sigma_{k-1}(\beta,b).</p>

<p><a id="X82C839D7794DDBCD" name="X82C839D7794DDBCD"></a></p>

<h4>3.2 <span class="Heading">Compatible elements</span></h4>

<p>This section is concerned with testing compatibility of two given elements (see <code class="func">AreCompatibleBallElements</code> (<a href="chap3.html#X84AA17C28591AC1F"><span class="RefLink">3.2-1</span></a>)) and finding an/all elements that is/are compatible with a given one (see <code class="func">CompatibleBallElement</code> (<a href="chap3.html#X8206962582F9C96A"><span class="RefLink">3.2-2</span></a>), <code class="func">CompatibilitySet</code> (<a href="chap3.html#X7ECC9E9982597E12"><span class="RefLink">3.2-3</span></a>)).</p>

<p><a id="X84AA17C28591AC1F" name="X84AA17C28591AC1F"></a></p>

<h5>3.2-1 AreCompatibleBallElements</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AreCompatibleBallElements</code>( <var class="Arg">d</var>, <var class="Arg">k</var>, <var class="Arg">aut1</var>, <var class="Arg">aut2</var>, <var class="Arg">dir</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: <code class="keyw">true</code> if <var class="Arg">aut1</var> and <var class="Arg">aut2</var> are compatible with each other in direction <var class="Arg">dir</var>, and <code class="keyw">false</code> otherwise.</p>

<p>The arguments of this method are a degree <var class="Arg">d</var> <span class="Math">\in\mathbb{N}_{\ge 3}</span>, a radius <var class="Arg">k</var> <span class="Math">\in\mathbb{N}</span>, two automorphisms <var class="Arg">aut1</var>, <var class="Arg">aut2</var> <span class="Math">\in\mathrm{Aut}(B_{d,k})</span>, and a direction <var class="Arg">dir</var> <span class="Math">\in</span><code class="code">[1..d]</code>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">AreCompatibleBallElements(3,1,(1,2),(1,2,3),1);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">AreCompatibleBallElements(3,1,(1,2),(1,2,3),2);</span>
false
</pre></div>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">a:=(1,3,5)(2,4,6);; a in AutBall(3,2);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">LocalAction(1,3,2,a,[]); LocalAction(1,3,2,a,[1]);</span>
(1,2,3)
(1,2)
<span class="GAPprompt">gap></span> <span class="GAPinput">b:=(1,4)(2,3);; b in AutBall(3,2);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">LocalAction(1,3,2,b,[]); LocalAction(1,3,2,b,[1]);</span>
(1,2)
(1,2,3)
<span class="GAPprompt">gap></span> <span class="GAPinput">AreCompatibleBallElements(3,2,a,b,1);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">AreCompatibleBallElements(3,2,a,b,3);</span>
false
</pre></div>

<p><a id="X8206962582F9C96A" name="X8206962582F9C96A"></a></p>

<h5>3.2-2 CompatibleBallElement</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CompatibleBallElement</code>( <var class="Arg">F</var>, <var class="Arg">aut</var>, <var class="Arg">dir</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: an element of <var class="Arg">F</var> that is compatible with <var class="Arg">aut</var> in direction <var class="Arg">dir</var> if one exists, and <code class="keyw">fail</code> otherwise.</p>

<p>The arguments of this method are a local action <var class="Arg">F</var> <span class="Math">\le\mathrm{Aut}(B_{d,k})</span>, an element <var class="Arg">aut</var> <span class="Math">\in</span> <var class="Arg">F</var>, and a direction <var class="Arg">dir</var> <span class="Math">\in</span><code class="code">[1..d]</code>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">mt:=RandomSource(IsMersenneTwister,1);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">a:=Random(mt,AutBall(5,1)); dir:=Random(mt,[1..5]);</span>
(1,2,5,4,3)
4
<span class="GAPprompt">gap></span> <span class="GAPinput">CompatibleBallElement(AutBall(5,1),a,dir);</span>
(1,2,5,4,3)
</pre></div>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">a:=(1,3,5)(2,4,6);; a in AutBall(3,2);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">CompatibleBallElement(AutBall(3,2),a,1);</span>
(1,4,2,3)
</pre></div>

<p><a id="X7ECC9E9982597E12" name="X7ECC9E9982597E12"></a></p>

<h5>3.2-3 <span class="Heading">CompatibilitySet</span></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CompatibilitySet</code>( <var class="Arg">F</var>, <var class="Arg">aut</var>, <var class="Arg">dir</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">‣ CompatibilitySet</code>( <var class="Arg">F</var>, <var class="Arg">aut</var>, <var class="Arg">dirs</var> )</td><td class="tdright">( operation )</td></tr></table></div>

<dl>
<dt><strong class="Mark">for the arguments <var class="Arg">F</var>, <var class="Arg">aut</var>, <var class="Arg">dir</var></strong></dt>
<dd><p>Returns: the list of elements of <var class="Arg">F</var> that are compatible with <var class="Arg">aut</var> in direction <var class="Arg">dir</var>.</p>

<p>The arguments of this method are a local action <var class="Arg">F</var> of <span class="Math">\le\mathrm{Aut}(B_{d,k})</span>, an automorphism <var class="Arg">aut</var> <span class="Math">\in F</span>, and a direction <var class="Arg">dir</var> <span class="Math">\in</span><code class="code">[1..d]</code>.</p>

</dd>
<dt><strong class="Mark">for the arguments <var class="Arg">F</var>, <var class="Arg">aut</var>, <var class="Arg">dirs</var></strong></dt>
<dd><p>Returns: the list of elements of <var class="Arg">F</var> that are compatible with <var class="Arg">aut</var> in all directions of <var class="Arg">dirs</var>.</p>

<p>The arguments of this method are a local action <var class="Arg">F</var> of <span class="Math">\le\mathrm{Aut}(B_{d,k})</span>, an automorphism <var class="Arg">aut</var> <span class="Math">\in F</span>, and a sublist of directions <var class="Arg">dirs</var> <span class="Math">\subseteq</span><code class="code">[1..d]</code>.</p>

</dd>
</dl>

<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">F:=LocalAction(4,1,TransitiveGroup(4,3));</span>
D(4)
<span class="GAPprompt">gap></span> <span class="GAPinput">G:=LocalAction(4,1,SymmetricGroup(4));</span>
Sym( [ 1 .. 4 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">aut:=(1,3);; aut in F;</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">CompatibilitySet(G,aut,1);</span>
RightCoset(Sym( [ 2 .. 4 ] ),(1,3))
<span class="GAPprompt">gap></span> <span class="GAPinput">CompatibilitySet(F,aut,1);</span>
RightCoset(Group([ (2,4) ]),(1,3))
<span class="GAPprompt">gap></span> <span class="GAPinput">CompatibilitySet(F,aut,[1,3]);</span>
RightCoset(Group([ (2,4) ]),(1,3))
<span class="GAPprompt">gap></span> <span class="GAPinput">CompatibilitySet(F,aut,[1,2]);</span>
RightCoset(Group(()),(1,3))
</pre></div>

<p><a id="X837E15117DB5B1DE" name="X837E15117DB5B1DE"></a></p>

<h5>3.2-4 AssembleAutomorphism</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AssembleAutomorphism</code>( <var class="Arg">d</var>, <var class="Arg">k</var>, <var class="Arg">auts</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: the automorphism <span class="Math">(</span><code class="code">aut</code><span class="Math">,(</span><var class="Arg">auts</var><span class="Math">[</span><code class="code">i</code><span class="Math">])_{i=1}^{d})</span> of <span class="Math">B_{d,k+1}</span>, where <code class="code">aut</code> is implicit in <span class="Math">(</span><var class="Arg">auts</var><span class="Math">[</span><code class="code">i</code><span class="Math">])_{i=1}^{d}</span>.</p>

<p>The arguments of this method are a degree <var class="Arg">d</var> <span class="Math">\in\mathbb{N}_{\ge 3}</span>, a radius <var class="Arg">k</var> <span class="Math">\in\mathbb{N}</span>, and a list <var class="Arg">auts</var> of <var class="Arg">d</var> automorphisms <span class="Math">(</span><var class="Arg">auts</var><span class="Math">[</span><code class="code">i</code><span class="Math">])_{i=1}^{d}</span> of <span class="Math">B_{d,k}</span> which comes from an element <span class="Math">(</span><code class="code">aut</code><span class="Math">,(</span><var class="Arg">auts</var><span class="Math">[</span><code class="code">i</code><span class="Math">])_{i=1}^{d})</span> of <span class="Math">\mathrm{Aut}(B_{d,k+1})</span>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">mt:=RandomSource(IsMersenneTwister,1);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">aut:=Random(mt,AutBall(3,2));</span>
(1,4,5,2,3,6)
<span class="GAPprompt">gap></span> <span class="GAPinput">auts:=[];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">for i in [1..3] do auts[i]:=CompatibleBallElement(AutBall(3,2),aut,i); od;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">auts;</span>
[ (1,4,6,2,3,5), (1,3,6,2,4,5), (1,5)(2,6) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">a:=AssembleAutomorphism(3,2,auts);</span>
(1,7,9,3,5,11)(2,8,10,4,6,12)
<span class="GAPprompt">gap></span> <span class="GAPinput">a in AutBall(3,3); </span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">LocalAction(2,3,3,a,[]);</span>
(1,4,5,2,3,6)
</pre></div>

<p><a id="X7967612B81C5074E" name="X7967612B81C5074E"></a></p>

<h4>3.3 <span class="Heading">Compatible subgroups</span></h4>

<p>Using the methods of Section <a href="chap3.html#X82C839D7794DDBCD"><span class="RefLink">3.2</span></a>, this section provides methods to test groups for the compatibility condition and search for compatible subgroups inside a given group, e.g. <span class="Math">\mathrm{Aut}(B_{d,k})</span>, or with a certain image under some projection.</p>

<p><a id="X79D1BC54814345D4" name="X79D1BC54814345D4"></a></p>

<h5>3.3-1 MaximalCompatibleSubgroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MaximalCompatibleSubgroup</code>( <var class="Arg">F</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: The local action <span class="Math">C(</span><var class="Arg">F</var><span class="Math">)\le\mathrm{Aut}(B_{d,k})</span>, which is the maximal compatible subgroup of <var class="Arg">F</var>.</p>

<p>The argument of this attribute is a local action <var class="Arg">F</var> <span class="Math">\le\mathrm{Aut}(B_{d,k})</span> (see <code class="func">IsLocalAction</code> (<a href="chap2.html#X7FCF15167D3A44B7"><span class="RefLink">2.1-1</span></a>)).</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">F:=LocalAction(3,1,Group((1,2)));</span>
Group([ (1,2) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">MaximalCompatibleSubgroup(F);</span>
Group([ (1,2) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">G:=LocalAction(3,2,Group((1,2)));</span>
Group([ (1,2) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">MaximalCompatibleSubgroup(G);</span>
Group(())
</pre></div>

<p><a id="X8302F28A85F1C4FE" name="X8302F28A85F1C4FE"></a></p>

<h5>3.3-2 SatisfiesC</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SatisfiesC</code>( <var class="Arg">F</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: <code class="keyw">true</code> if <var class="Arg">F</var> satisfies the compatibility condition (C), and <code class="keyw">false</code> otherwise.</p>

<p>The argument of this property is a local action <var class="Arg">F</var> <span class="Math">\le\mathrm{Aut}(B_{d,k})</span> (see <code class="func">IsLocalAction</code> (<a href="chap2.html#X7FCF15167D3A44B7"><span class="RefLink">2.1-1</span></a>)).</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">D:=LocalActionDelta(3,SymmetricGroup(3));</span>
Group([ (1,3,6)(2,4,5), (1,3)(2,4), (1,2)(3,4)(5,6) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">SatisfiesC(D);</span>
true
</pre></div>

<p><a id="X7FF0262182F3B9E8" name="X7FF0262182F3B9E8"></a></p>

<h5>3.3-3 CompatibleSubgroups</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CompatibleSubgroups</code>( <var class="Arg">F</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: the list of all compatible subgroups of <var class="Arg">F</var>.</p>

<p>The argument of this method is a local action <var class="Arg">F</var> <span class="Math">\le\mathrm{Aut}(B_{d,k})</span>. This method calls <code class="code">AllSubgroups</code> on <span class="Math">F</span> and is therefore slow. Use for instructional purposes on small examples only, and use <code class="func">ConjugacyClassRepsCompatibleSubgroups</code> (<a href="chap3.html#X7CEBD97183C3399E"><span class="RefLink">3.3-4</span></a>) or <code class="func">ConjugacyClassRepsCompatibleGroupsWithProjection</code> (<a href="chap3.html#X84F140AC7AD8D9EE"><span class="RefLink">3.3-5</span></a>) for computations.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G:=LocalActionGamma(3,SymmetricGroup(3));</span>
Group([ (1,4,5)(2,3,6), (1,3)(2,4)(5,6) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">list:=CompatibleSubgroups(G);</span>
[ Group(()), Group([ (1,2)(3,5)(4,6) ]), Group([ (1,3)(2,4)(5,6) ]), 
  Group([ (1,6)(2,5)(3,4) ]), Group([ (1,4,5)(2,3,6) ]), 
  Group([ (1,4,5)(2,3,6), (1,3)(2,4)(5,6) ]) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Size(list);</span>
6
<span class="GAPprompt">gap></span> <span class="GAPinput">Size(AllSubgroups(SymmetricGroup(3)));</span>
6
</pre></div>

<p><a id="X7CEBD97183C3399E" name="X7CEBD97183C3399E"></a></p>

<h5>3.3-4 ConjugacyClassRepsCompatibleSubgroups</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ConjugacyClassRepsCompatibleSubgroups</code>( <var class="Arg">F</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a list of compatible representatives of conjugacy classes of <var class="Arg">F</var> that contain a compatible subgroup.</p>

<p>The argument of this method is a local action <var class="Arg">F</var> of <span class="Math">\mathrm{Aut}(B_{d,k})</span>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">ConjugacyClassRepsCompatibleSubgroups(AutBall(3,2));</span>
[ Group(()), Group([ (1,2)(3,5)(4,6) ]), Group([ (1,4,5)(2,3,6) ]), 
  Group([ (3,5)(4,6), (1,2) ]), Group([ (1,2)(3,5)(4,6), (1,3,6)(2,4,5) ]), 
  Group([ (3,5)(4,6), (1,3,5)(2,4,6), (1,2)(3,4)(5,6) ]), 
  Group([ (1,2)(3,5)(4,6), (1,3,5)(2,4,6), (1,2)(5,6), (1,2)(3,4) ]), 
  Group([ (3,5)(4,6), (1,3,5)(2,4,6), (1,2)(5,6), (1,2)(3,4) ]), 
  Group([ (5,6), (3,4), (1,2), (1,3,5)(2,4,6), (3,5)(4,6) ]) ]
</pre></div>

<p><a id="X84F140AC7AD8D9EE" name="X84F140AC7AD8D9EE"></a></p>

<h5>3.3-5 ConjugacyClassRepsCompatibleGroupsWithProjection</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ConjugacyClassRepsCompatibleGroupsWithProjection</code>( <var class="Arg">l</var>, <var class="Arg">F</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: a list of compatible representatives of conjugacy classes of <span class="Math">\mathrm{Aut}(B_{d,l})</span> that contain a compatible group which projects to <var class="Arg">F</var> <span class="Math">\le\mathrm{Aut}(B_{d,r})</span>.</p>

<p>The arguments of this method are a radius <var class="Arg">l</var> <span class="Math">\in\mathbb{N}</span>, and a local action <var class="Arg">F</var> <span class="Math">\le\mathrm{Aut}(B_{d,k})</span> for some <span class="Math">k\le l</span>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">S3:=LocalAction(3,1,SymmetricGroup(3));</span>
Sym( [ 1 .. 3 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">ConjugacyClassRepsCompatibleGroupsWithProjection(2,S3);</span>
[ Group([ (1,2)(3,5)(4,6), (1,4,5)(2,3,6) ]), 
  Group([ (1,2)(3,4)(5,6), (1,2)(3,5)(4,6), (1,4,5)(2,3,6) ]), 
  Group([ (3,4)(5,6), (1,2)(3,4), (1,4,5)(2,3,6), (3,5,4,6) ]), 
  Group([ (3,4)(5,6), (1,2)(3,4), (1,4,5)(2,3,6), (3,5)(4,6) ]), 
  Group([ (3,4)(5,6), (1,2)(3,4), (1,4,5)(2,3,6), (5,6), (3,5,4,6) ]) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">A3:=LocalAction(3,1,AlternatingGroup(3));</span>
Alt( [ 1 .. 3 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">ConjugacyClassRepsCompatibleGroupsWithProjection(2,A3);</span>
[ Group([ (1,4,5)(2,3,6) ]) ]
</pre></div>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">F:=SymmetricGroup(3);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">rho:=SignHomomorphism(F);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">H1:=LocalActionPi(2,3,F,rho,[0,1]);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">H2:=LocalActionPi(2,3,F,rho,[1]);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Size(ConjugacyClassRepsCompatibleGroupsWithProjection(3,H1));</span>
2
<span class="GAPprompt">gap></span> <span class="GAPinput">Size(ConjugacyClassRepsCompatibleGroupsWithProjection(3,H2));</span>
4
</pre></div>


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