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<div class="chlinktop"><span class="chlink1">Goto Chapter: </span><a href="chap0_mj.html">Top</a> <a href="chap1_mj.html">1</a> <a href="chap2_mj.html">2</a> <a href="chap3_mj.html">3</a> <a href="chap4_mj.html">4</a> <a href="chap5_mj.html">5</a> <a href="chap6_mj.html">6</a> <a href="chap7_mj.html">7</a> <a href="chap8_mj.html">8</a> <a href="chap9_mj.html">9</a> <a href="chapA_mj.html">A</a> <a href="chapB_mj.html">B</a> <a href="chapBib_mj.html">Bib</a> <a href="chapInd_mj.html">Ind</a> </div>
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<div class="chlinkprevnexttop"> <a href="chap0_mj.html">[Top of Book]</a> <a href="chap0_mj.html#contents">[Contents]</a> <a href="chap7_mj.html">[Previous Chapter]</a> <a href="chap9_mj.html">[Next Chapter]</a> </div>
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<div class="ChapSects"><a href="chap8_mj.html#X85AFC9C47FD3C03F">8 <span class="Heading">Specific Methods</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap8_mj.html#X7990F2F880E717EE">8.1 <span class="Heading">Core Methods for Bol Loops</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X8664CA927DD73DBE">8.1-1 <span class="Heading">AssociatedLeftBruckLoop and AssociatedRightBruckLoop</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X82FC16F386CE11F1">8.1-2 IsExactGroupFactorization</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X7DCA64807F899127">8.1-3 RightBolLoopByExactGroupFactorization</a></span>
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</div></div>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap8_mj.html#X819F82737C2A860D">8.2 <span class="Heading">Moufang Modifications</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X7B3165C083709831">8.2-1 LoopByCyclicModification</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X7D7717C587BC2D1E">8.2-2 LoopByDihedralModification</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X7CC6CDB786E9BBA0">8.2-3 LoopMG2</a></span>
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</div></div>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap8_mj.html#X83E73A767D79FAFD">8.3 <span class="Heading">Triality for Moufang Loops</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X7DB4DE647F6F56F0">8.3-1 TrialityPermGroup</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X82CC977085DFDFE8">8.3-2 TrialityPcGroup</a></span>
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</div></div>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap8_mj.html#X841ED66B8084AA73">8.4 <span class="Heading">Realizing Groups as Multiplication Groups of Loops</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X804F40087DD1225D">8.4-1 AllLoopTablesInGroup</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X7854C8E382DC8E8B">8.4-2 AllProperLoopTablesInGroup</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X7BFFC66A824BA6AA">8.4-3 OneLoopTableInGroup</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X84C5A76585B335FF">8.4-4 OneProperLoopTableInGroup</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X7E5F1C2879358EEF">8.4-5 AllLoopsWithMltGroup</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X8266DE05824226E6">8.4-6 OneLoopWithMltGroup</a></span>
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</div></div>
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</div>
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<h3>8 <span class="Heading">Specific Methods</span></h3>
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<p>This chapter describes methods of <strong class="pkg">LOOPS</strong> that apply to specific classes of loops, mostly Bol and Moufang loops.</p>
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<p><a id="X7990F2F880E717EE" name="X7990F2F880E717EE"></a></p>
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<h4>8.1 <span class="Heading">Core Methods for Bol Loops</span></h4>
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<p><a id="X8664CA927DD73DBE" name="X8664CA927DD73DBE"></a></p>
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<h5>8.1-1 <span class="Heading">AssociatedLeftBruckLoop and AssociatedRightBruckLoop</span></h5>
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<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AssociatedLeftBruckLoop</code>( <var class="Arg">Q</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
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<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AssociatedRightBruckLoop</code>( <var class="Arg">Q</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
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<p>Returns: The left (resp. right) Bruck loop associated with a uniquely 2-divisible left (resp. right) Bol loop <var class="Arg">Q</var>.</p>
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<p>Let <span class="SimpleMath">\(Q\)</span> be a left Bol loop such that the mapping <span class="SimpleMath">\(x\mapsto x^2\)</span> is a permutation of <span class="SimpleMath">\(Q\)</span>. Define a new operation <span class="SimpleMath">\(*\)</span> on <span class="SimpleMath">\(Q\)</span> by <span class="SimpleMath">\(x*y =(x(y^2x))^{1/2}\)</span>. Then <span class="SimpleMath">\((Q,*)\)</span> is a left Bruck loop, called the <em>associated left Bruck loop</em>. (In fact, Bruck used the isomorphic operation <span class="SimpleMath">\(x*y = x^{1/2}(yx^{1/2})\)</span> instead. Our approach is more natural in the sense that the left Bruck loop associated with a left Bruck loop is identical to the original loop.) Associated right Bruck loops are defined dually.</p>
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<p><a id="X82FC16F386CE11F1" name="X82FC16F386CE11F1"></a></p>
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<h5>8.1-2 IsExactGroupFactorization</h5>
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<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsExactGroupFactorization</code>( <var class="Arg">G</var>, <var class="Arg">H1</var>, <var class="Arg">H2</var> )</td><td class="tdright">( operation )</td></tr></table></div>
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<p>Returns: <code class="code">true</code> if (<var class="Arg">G</var>, <var class="Arg">H1</var>, <var class="Arg">H2</var>) is an exact group factorization.</p>
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<p>Many right Bol loops can be constructed from exact group factorizations. The triple <span class="SimpleMath">\((G,H_1,H_2)\)</span> is an <em>exact group factorization</em> if <span class="SimpleMath">\(H_1\)</span>, <span class="SimpleMath">\(H_2\)</span> are subgroups of <span class="SimpleMath">\(G\)</span> such that <span class="SimpleMath">\(H_1H_2=G\)</span> and <span class="SimpleMath">\(H_1\cap H_2=1\)</span>.</p>
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<p><a id="X7DCA64807F899127" name="X7DCA64807F899127"></a></p>
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<h5>8.1-3 RightBolLoopByExactGroupFactorization</h5>
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<p>If <span class="SimpleMath">\((G,H_1,H_2)\)</span> is an exact group factorization then <span class="SimpleMath">\((G\times G, H_1\times H_2, T)\)</span> with <span class="SimpleMath">\(T=\{(x,x^{-1})| x\in G\}\)</span> is a loop folder that gives rise to a right Bol loop.</p>
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<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RightBolLoopByExactGroupFactorization</code>( <var class="Arg">arg</var> )</td><td class="tdright">( function )</td></tr></table></div>
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<p>Returns: The right Bol loop constructed from an exact group factorization. The argument <var class="Arg">arg</var> can either be an exact group factorization <code class="code">[G,H1,H2]</code>, or the tuple <code class="code">[G,H]</code>, where <code class="code">H</code> is a regular subgroup of <code class="code">G</code>. We also allow <var class="Arg">arg</var> to be separate entries rather than a list of entries.</p>
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<p><a id="X819F82737C2A860D" name="X819F82737C2A860D"></a></p>
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<h4>8.2 <span class="Heading">Moufang Modifications</span></h4>
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<p>Drápal <a href="chapBib_mj.html#biBDrapalCD">[Drá03]</a> described two prominent families of extensions of Moufang loops. It turns out that these extensions suffice to obtain all nonassociative Moufang loops of order at most 64 if one starts with so-called Chein loops. We call the two constructions <em>Moufang modifications</em>. The library of Moufang loops included in <strong class="pkg">LOOPS</strong> is based on Moufang modifications. See <a href="chapBib_mj.html#biBDrVo">[DV06]</a> for details.</p>
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<p><a id="X7B3165C083709831" name="X7B3165C083709831"></a></p>
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<h5>8.2-1 LoopByCyclicModification</h5>
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<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LoopByCyclicModification</code>( <var class="Arg">Q</var>, <var class="Arg">S</var>, <var class="Arg">a</var>, <var class="Arg">h</var> )</td><td class="tdright">( function )</td></tr></table></div>
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<p>Returns: The cyclic modification of a Moufang loop <var class="Arg">Q</var> obtained from <var class="Arg">S</var>, <var class="Arg">a</var><span class="SimpleMath">\(=\alpha\)</span> and <var class="Arg">h</var> described below.</p>
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<p>Assume that <span class="SimpleMath">\(Q\)</span> is a Moufang loop with a normal subloop <span class="SimpleMath">\(S\)</span> such that <span class="SimpleMath">\(Q/S\)</span> is a cyclic group of order <span class="SimpleMath">\(2m\)</span>. Let <span class="SimpleMath">\(h\in S\cap Z(L)\)</span>. Let <span class="SimpleMath">\(\alpha\)</span> be a generator of <span class="SimpleMath">\(Q/S\)</span> and write <span class="SimpleMath">\(Q = \bigcup_{i\in M} \alpha^i\)</span>, where <span class="SimpleMath">\(M=\{-m+1\)</span>, <span class="SimpleMath">\(\dots\)</span>, <span class="SimpleMath">\(m\}\)</span>. Let <span class="SimpleMath">\(\sigma:\mathbb{Z}\to M\)</span> be defined by <span class="SimpleMath">\(\sigma(i)=0\)</span> if <span class="SimpleMath">\(i\in M\)</span>, <span class="SimpleMath">\(\sigma(i)=1\)</span> if <span class="SimpleMath">\(i>m\)</span>, and <span class="SimpleMath">\(\sigma(i)=-1\)</span> if <span class="SimpleMath">\(i<-m+1\)</span>. Introduce a new multiplication <span class="SimpleMath">\(*\)</span> on <span class="SimpleMath">\(Q\)</span> by <span class="SimpleMath">\(x*y = xyh^{\sigma(i+j)}\)</span>, where <span class="SimpleMath">\(x\in \alpha^i\)</span>, <span class="SimpleMath">\(y\in\alpha^j\)</span>, <span class="SimpleMath">\(i\in M\)</span> and <span class="SimpleMath">\(j\in M\)</span>. Then <span class="SimpleMath">\((Q,*)\)</span> is a Moufang loop, a <em>cyclic modification</em> of <span class="SimpleMath">\(Q\)</span>.</p>
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<p><a id="X7D7717C587BC2D1E" name="X7D7717C587BC2D1E"></a></p>
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<h5>8.2-2 LoopByDihedralModification</h5>
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<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LoopByDihedralModification</code>( <var class="Arg">Q</var>, <var class="Arg">S</var>, <var class="Arg">e</var>, <var class="Arg">f</var>, <var class="Arg">h</var> )</td><td class="tdright">( function )</td></tr></table></div>
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<p>Returns: The dihedral modification of a Moufang loop <var class="Arg">Q</var> obtained from <var class="Arg">S</var>, <var class="Arg">e</var>, <var class="Arg">f</var> and <var class="Arg">h</var> as described below.</p>
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<p>Let <span class="SimpleMath">\(Q\)</span> be a Moufang loop with a normal subloop <span class="SimpleMath">\(S\)</span> such that <span class="SimpleMath">\(Q/S\)</span> is a dihedral group of order <span class="SimpleMath">\(4m\)</span>, with <span class="SimpleMath">\(m\ge 1\)</span>. Let <span class="SimpleMath">\(M\)</span> and <span class="SimpleMath">\(\sigma\)</span> be defined as in the cyclic case. Let <span class="SimpleMath">\(\beta\)</span>, <span class="SimpleMath">\(\gamma\)</span> be two involutions of <span class="SimpleMath">\(Q/S\)</span> such that <span class="SimpleMath">\(\alpha=\beta\gamma\)</span> generates a cyclic subgroup of <span class="SimpleMath">\(Q/S\)</span> of order <span class="SimpleMath">\(2m\)</span>. Let <span class="SimpleMath">\(e\in\beta\)</span> and <span class="SimpleMath">\(f\in\gamma\)</span> be arbitrary. Then <span class="SimpleMath">\(Q\)</span> can be written as a disjoint union <span class="SimpleMath">\(Q=\bigcup_{i\in M}(\alpha^i\cup e\alpha^i)\)</span>, and also <span class="SimpleMath">\(Q=\bigcup_{i\in M}(\alpha^i\cup \alpha^if)\)</span>. Let <span class="SimpleMath">\(G_0=\bigcup_{i\in M}\alpha^i\)</span>, and <span class="SimpleMath">\(G_1=L\setminus G_0\)</span>. Let <span class="SimpleMath">\(h\in S\cap N(L)\cap Z(G_0)\)</span>. Introduce a new multiplication <span class="SimpleMath">\(*\)</span> on <span class="SimpleMath">\(Q\)</span> by <span class="SimpleMath">\(x*y = xyh^{(-1)^r\sigma(i+j)}\)</span>, where <span class="SimpleMath">\(x\in\alpha^i\cup e\alpha^i\)</span>, <span class="SimpleMath">\(y\in\alpha^j\cup \alpha^jf\)</span>, <span class="SimpleMath">\(i\in M\)</span>, <span class="SimpleMath">\(j\in M\)</span>, <span class="SimpleMath">\(y\in G_r\)</span> and <span class="SimpleMath">\(r\in\{0,1\}\)</span>. Then <span class="SimpleMath">\((Q,*)\)</span> is a Moufang loop, a <em>dihedral modification</em> of <span class="SimpleMath">\(Q\)</span>.</p>
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<p><a id="X7CC6CDB786E9BBA0" name="X7CC6CDB786E9BBA0"></a></p>
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<h5>8.2-3 LoopMG2</h5>
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<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LoopMG2</code>( <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
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<p>Returns: The Chein loop constructed from a group <var class="Arg">G</var>.</p>
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<p>Let <span class="SimpleMath">\(G\)</span> be a group. Let <span class="SimpleMath">\(\overline{G}=\{\overline{g}|g\in G\}\)</span> be a disjoint copy of elements of <span class="SimpleMath">\(G\)</span>. Define multiplication <span class="SimpleMath">\(*\)</span> on <span class="SimpleMath">\(Q=G\cup \overline{G}\)</span> by <span class="SimpleMath">\(g*h = gh\)</span>, <span class="SimpleMath">\(g*\overline{h}=\overline{hg}\)</span>, <span class="SimpleMath">\(\overline{g}*h = \overline{gh^{-1}}\)</span> and <span class="SimpleMath">\(\overline{g}*\overline{h}=h^{-1}g\)</span>, where <span class="SimpleMath">\(g\)</span>, <span class="SimpleMath">\(h\in G\)</span>. Then <span class="SimpleMath">\((Q,*)=M(G,2)\)</span> is a so-called <em>Chein loop</em>, which is always a Moufang loop, and it is associative if and only if <span class="SimpleMath">\(G\)</span> is commutative.</p>
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<p><a id="X83E73A767D79FAFD" name="X83E73A767D79FAFD"></a></p>
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<h4>8.3 <span class="Heading">Triality for Moufang Loops</span></h4>
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<p>Let <span class="SimpleMath">\(G\)</span> be a group and <span class="SimpleMath">\(\sigma\)</span>, <span class="SimpleMath">\(\rho\)</span> be automorphisms of <span class="SimpleMath">\(G\)</span> satisfying <span class="SimpleMath">\(\sigma^2 = \rho^3 = (\sigma \rho)^2 = 1\)</span>. Below we write automorphisms as exponents and <span class="SimpleMath">\([g,\sigma]\)</span> for <span class="SimpleMath">\(g^{-1}g^\sigma\)</span>. We say that the triple <span class="SimpleMath">\((G,\rho,\sigma)\)</span> is a <em>group with triality</em> if <span class="SimpleMath">\([g, \sigma] [g,\sigma]^\rho [g,\sigma]^{\rho^2} =1\)</span> holds for all <span class="SimpleMath">\(g \in G\)</span>. It is known that one can associate a group with triality <span class="SimpleMath">\((G,\rho,\sigma)\)</span> in a canonical way with a Moufang loop <span class="SimpleMath">\(Q\)</span>. See <a href="chapBib_mj.html#biBNaVo2003">[NV03]</a> for more details.</p>
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<p>For any Moufang loop <span class="SimpleMath">\(Q\)</span>, we can calculate the triality group as a permutation group acting on <span class="SimpleMath">\(3|Q|\)</span> points. If the multiplication group of <span class="SimpleMath">\(Q\)</span> is polycyclic, then we can also represent the triality group as a pc group. In both cases, the automorphisms <span class="SimpleMath">\(\sigma\)</span> and <span class="SimpleMath">\(\rho\)</span> are in the same family as the elements of <span class="SimpleMath">\(G\)</span>.</p>
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<p><a id="X7DB4DE647F6F56F0" name="X7DB4DE647F6F56F0"></a></p>
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<h5>8.3-1 TrialityPermGroup</h5>
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<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TrialityPermGroup</code>( <var class="Arg">Q</var> )</td><td class="tdright">( function )</td></tr></table></div>
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<p>Returns: A record with components <code class="code">G</code>, <code class="code">rho</code>, <code class="code">sigma</code>, where <code class="code">G</code> is the canonical group with triality associated with a Moufang loop <var class="Arg">Q</var>, and <code class="code">rho</code>, <code class="code">sigma</code> are the corresponding triality automorphisms.</p>
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<p><a id="X82CC977085DFDFE8" name="X82CC977085DFDFE8"></a></p>
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<h5>8.3-2 TrialityPcGroup</h5>
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<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TrialityPcGroup</code>( <var class="Arg">Q</var> )</td><td class="tdright">( function )</td></tr></table></div>
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<p>This is a variation of <code class="code">TrialityPermGroup</code> in which <code class="code">G</code> is returned as a pc group.</p>
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<p><a id="X841ED66B8084AA73" name="X841ED66B8084AA73"></a></p>
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<h4>8.4 <span class="Heading">Realizing Groups as Multiplication Groups of Loops</span></h4>
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<p>It is difficult to determine which groups can occur as multiplication groups of loops.</p>
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<p>The following operations search for loops whose multiplication groups are contained within a specified transitive permutation group <var class="Arg">G</var>. In all these operations, one can speed up the search by increasing the optional argument <var class="Arg">depth</var>, the price being a much higher memory consumption. The argument <var class="Arg">depth</var> is optimally chosen if in the permutation group <var class="Arg">G</var> there are not many permutations fixing <var class="Arg">depth</var> elements. It is safe to omit the argument or set it equal to 2.</p>
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<p>The optional argument <var class="Arg">infolevel</var> determines the amount of information displayed during the search. With <code class="code"><var class="Arg">infolevel</var>=0</code>, no information is provided. With <code class="code"><var class="Arg">infolevel</var>=1</code>, you get some information on timing and hits. With <code class="code"><var class="Arg">infolevel</var>=2</code>, the results are printed as well.</p>
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<p><a id="X804F40087DD1225D" name="X804F40087DD1225D"></a></p>
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<h5>8.4-1 AllLoopTablesInGroup</h5>
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<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AllLoopTablesInGroup</code>( <var class="Arg">G</var>[, <var class="Arg">depth</var>[, <var class="Arg">infolevel</var>]] )</td><td class="tdright">( operation )</td></tr></table></div>
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<p>Returns: All Cayley tables of loops whose multiplication group is contained in the transitive permutation group <var class="Arg">G</var>.</p>
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<p><a id="X7854C8E382DC8E8B" name="X7854C8E382DC8E8B"></a></p>
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<h5>8.4-2 AllProperLoopTablesInGroup</h5>
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<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AllProperLoopTablesInGroup</code>( <var class="Arg">G</var>[, <var class="Arg">depth</var>[, <var class="Arg">infolevel</var>]] )</td><td class="tdright">( operation )</td></tr></table></div>
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<p>Returns: All Cayley tables of nonassociative loops whose multiplication group is contained in the transitive permutation group <var class="Arg">G</var>.</p>
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<p><a id="X7BFFC66A824BA6AA" name="X7BFFC66A824BA6AA"></a></p>
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<h5>8.4-3 OneLoopTableInGroup</h5>
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<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ OneLoopTableInGroup</code>( <var class="Arg">G</var>[, <var class="Arg">depth</var>[, <var class="Arg">infolevel</var>]] )</td><td class="tdright">( operation )</td></tr></table></div>
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<p>Returns: A Cayley table of a loop whose multiplication group is contained in the transitive permutation group <var class="Arg">G</var>.</p>
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<p><a id="X84C5A76585B335FF" name="X84C5A76585B335FF"></a></p>
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<h5>8.4-4 OneProperLoopTableInGroup</h5>
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<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ OneProperLoopTableInGroup</code>( <var class="Arg">G</var>[, <var class="Arg">depth</var>[, <var class="Arg">infolevel</var>]] )</td><td class="tdright">( operation )</td></tr></table></div>
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<p>Returns: A Cayley table of a nonassociative loop whose multiplication group is contained in the transitive permutation group <var class="Arg">G</var>.</p>
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<p><a id="X7E5F1C2879358EEF" name="X7E5F1C2879358EEF"></a></p>
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<h5>8.4-5 AllLoopsWithMltGroup</h5>
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<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AllLoopsWithMltGroup</code>( <var class="Arg">G</var>[, <var class="Arg">depth</var>[, <var class="Arg">infolevel</var>]] )</td><td class="tdright">( operation )</td></tr></table></div>
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<p>Returns: A list of all loops (given as sections) whose multiplication group is equal to the transitive permutation group <var class="Arg">G</var>.</p>
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<p><a id="X8266DE05824226E6" name="X8266DE05824226E6"></a></p>
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<h5>8.4-6 OneLoopWithMltGroup</h5>
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<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ OneLoopWithMltGroup</code>( <var class="Arg">G</var>[, <var class="Arg">depth</var>[, <var class="Arg">infolevel</var>]] )</td><td class="tdright">( operation )</td></tr></table></div>
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<p>Returns: One loop (given as a section) whose multiplication group is equal to the transitive permutation group <var class="Arg">G</var>.</p>
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<div class="example"><pre>
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<span class="GAPprompt">gap></span> <span class="GAPinput">g:=PGL(3,3);
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</span>
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Group([ (6,7)(8,11)(9,13)(10,12), (1,2,5,7,13,3,8,6,10,9,12,4,11) ])
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<span class="GAPprompt">gap></span> <span class="GAPinput">a:=AllLoopTablesInGroup(g,3,0);; Size(a);
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</span>
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56
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<span class="GAPprompt">gap></span> <span class="GAPinput">a:=AllLoopsWithMltGroup(g,3,0);; Size(a);
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</span>
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52
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</pre></div>
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