<p>The action of a group G on the coset space of a subgroup gives us a transitive permutation representation of the group. Whenever the subgroup is core-free, we have that the action of G on the coset space of the subgroup will be faithful. Moreover, the stabilizer of a point on a faithful transitive permutation representation of G will always be a core-free subgroup.</p>
<p>For a finite group <var class="Arg">G</var>, <var class="Arg">FaithfulTransitivePermutationRepresentations</var> returns a list of a faithful transitive permutation representation of <varclass="Arg">G</var> for each degree. If <var class="Arg">all_ftpr</var> is true, then it will return a list of all faithful transitive permutation representations, up to conjugacy equivalence.</p>
<h4>3.2 <span class="Heading">Faithful Transitive Permutation Representation of Minimal Degree</span></h4>
<p>To complement the already existing functions in GAP <var class="Arg">MinimalFaithfulPermutationDegree</var> and <var class="Arg">MinimalFaithfulPermutationRepresentation</var>, the following functions to retrieve the <var class="Arg">MinimalFaithfulTransitivePermutationRepresentation</var> and <var class="Arg">MinimalFaithfulTransitivePermutationDegree</var>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MinimalFaithfulTransitivePermutationRepresentation</code>( <var class="Arg">G</var>[, <var class="Arg">all_minimal_ftpr</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: an isomorphism (or a list of isomorphisms)</p>
<p>For a finite group <var class="Arg">G</var>, <var class="Arg">MinimalFaithfulTransitivePermutationRepresentation</var> returns an isomorphism of <var class="Arg">G</var> into the symmetric group of minimal degree acting transitively on its domain. If <var class="Arg">all_minimal_ftpr</var> is set as <var class="Arg">true</var>, then it returns a list of all isomorphisms <var class="Arg">G</var> into the symmetric group of minimal degree.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MinimalFaithfulTransitivePermutationDegree</code>( <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: an integer</p>
<p>For a finite group <var class="Arg">G</var>, <var class="Arg">MinimalFaithfulTransitivePermutationDegree</var> returns the least positive integer n such that <var class="Arg">G</var> is isomorphic to a subgroup of the symmetric group of degree n acting transitively on its domain.</p>
<h4>3.3 <span class="Heading">Faithful Transitive Permutation Representation of given Degree</span></h4>
<p>To obtain a faithful transitive permutation Representation of a specific degree, the following function <var class="Arg">FaithfulTransitivePermutationRepresentationsOfDegree</var> can be used.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FaithfulTransitivePermutationRepresentationsOfDegree</code>( <var class="Arg">G</var>, <var class="Arg">d</var>[, <var class="Arg">all_ftpr_of_given_degree</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: an isomorphism (or a list of isomorphisms)</p>
<p>For a finite group <var class="Arg">G</var>, <var class="Arg">FaithfulTransitivePermutationRepresentationsOfDegree</var> returns one isomorphism of <var class="Arg">G</var> into the symmetric group of degree <var class="Arg">d</var> acting transitively on its domain. If <var class="Arg">all_ftpr_of_given_degree</var> is set as <var class="Arg">true</var>, then it returns a list of all isomorphisms <var class="Arg">G</var> into the symmetric group of degree <var class="Arg">d</var>, up to conjugacy equivalence.</p>
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