  
  [1X3 [33X[0;0YFaithful Transitive Permutation Representations[133X[101X
  
  [33X[0;0YThe  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.[133X
  
  
  [1X3.1 [33X[0;0YObtaining Faithful Transitive Permutation Representations[133X[101X
  
  [1X3.1-1 FaithfulTransitivePermutationRepresentations[101X
  
  [33X[1;0Y[29X[2XFaithfulTransitivePermutationRepresentations[102X( [3XG[103X[, [3Xall_ftpr[103X] ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya list[133X
  
  [33X[0;0YFor a finite group [3XG[103X, [3XFaithfulTransitivePermutationRepresentations[103X returns a
  list  of  a  faithful  transitive  permutation  representation of [3XG[103X for each
  degree.  If  [3Xall_ftpr[103X  is  true,  then it will return a list of all faithful
  transitive permutation representations, up to conjugacy equivalence.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xsp := SymplecticGroup(4,2);;[127X[104X
    [4X[25Xgap>[125X [27XCoreFreeDegrees(sp);[127X[104X
    [4X[28X[ 6, 10, 12, 15, 20, 30, 36, 40, 45, 60, 72, 80, 90, 120, 144, 180, 240, 360, [128X[104X
    [4X[28X  720 ][128X[104X
    [4X[25Xgap>[125X [27Xftprs := FaithfulTransitivePermutationRepresentations(sp);; [127X[104X
    [4X[25Xgap>[125X [27XSize(ftprs);[127X[104X
    [4X[28X19[128X[104X
    [4X[25Xgap>[125X [27Xall_ftprs := FaithfulTransitivePermutationRepresentations(sp,true);; [127X[104X
    [4X[25Xgap>[125X [27XSize(all_ftprs);[127X[104X
    [4X[28X54[128X[104X
  [4X[32X[104X
  
  
  [1X3.2 [33X[0;0YFaithful Transitive Permutation Representation of Minimal Degree[133X[101X
  
  [33X[0;0YTo     complement     the     already     existing    functions    in    GAP
  [3XMinimalFaithfulPermutationDegree[103X                                         and
  [3XMinimalFaithfulPermutationRepresentation[103X,   the   following   functions   to
  retrieve    the    [3XMinimalFaithfulTransitivePermutationRepresentation[103X    and
  [3XMinimalFaithfulTransitivePermutationDegree[103X.[133X
  
  [1X3.2-1 MinimalFaithfulTransitivePermutationRepresentation[101X
  
  [33X[1;0Y[29X[2XMinimalFaithfulTransitivePermutationRepresentation[102X( [3XG[103X[, [3Xall_minimal_ftpr[103X] ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Yan isomorphism (or a list of isomorphisms)[133X
  
  [33X[0;0YFor  a  finite  group  [3XG[103X, [3XMinimalFaithfulTransitivePermutationRepresentation[103X
  returns  an  isomorphism  of  [3XG[103X  into  the symmetric group of minimal degree
  acting  transitively on its domain. If [3Xall_minimal_ftpr[103X is set as [3Xtrue[103X, then
  it  returns a list of all isomorphisms [3XG[103X into the symmetric group of minimal
  degree.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xsp := SymplecticGroup(4,2);;[127X[104X
    [4X[25Xgap>[125X [27Xmin_ftpr := MinimalFaithfulTransitivePermutationRepresentation(sp);[127X[104X
    [4X[28XCompositionMapping( <action epimorphism>, <action isomorphism> )[128X[104X
    [4X[25Xgap>[125X [27Xmin_ftprs := MinimalFaithfulTransitivePermutationRepresentation(sp,true);[127X[104X
    [4X[28X[ CompositionMapping( <action epimorphism>, <action isomorphism> ), [128X[104X
    [4X[28X  CompositionMapping( <action epimorphism>, <action isomorphism> ) ][128X[104X
  [4X[32X[104X
  
  [1X3.2-2 MinimalFaithfulTransitivePermutationDegree[101X
  
  [33X[1;0Y[29X[2XMinimalFaithfulTransitivePermutationDegree[102X( [3XG[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10Yan integer[133X
  
  [33X[0;0YFor a finite group [3XG[103X, [3XMinimalFaithfulTransitivePermutationDegree[103X returns the
  least  positive  integer  n  such  that [3XG[103X is isomorphic to a subgroup of the
  symmetric group of degree n acting transitively on its domain.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xsp := SymplecticGroup(4,2);; g:=SimpleGroup("PSL",3,5);;[127X[104X
    [4X[25Xgap>[125X [27XMinimalFaithfulTransitivePermutationDegree(sp);[127X[104X
    [4X[28X6[128X[104X
    [4X[25Xgap>[125X [27XMinimalFaithfulTransitivePermutationDegree(g);[127X[104X
    [4X[28X31[128X[104X
  [4X[32X[104X
  
  
  [1X3.3 [33X[0;0YFaithful Transitive Permutation Representation of given Degree[133X[101X
  
  [33X[0;0YTo  obtain  a  faithful  transitive permutation Representation of a specific
  degree,                 the                following                function
  [3XFaithfulTransitivePermutationRepresentationsOfDegree[103X can be used.[133X
  
  [1X3.3-1 FaithfulTransitivePermutationRepresentationsOfDegree[101X
  
  [33X[1;0Y[29X[2XFaithfulTransitivePermutationRepresentationsOfDegree[102X( [3XG[103X, [3Xd[103X[, [3Xall_ftpr_of_given_degree[103X] ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Yan isomorphism (or a list of isomorphisms)[133X
  
  [33X[0;0YFor  a  finite group [3XG[103X, [3XFaithfulTransitivePermutationRepresentationsOfDegree[103X
  returns  one  isomorphism  of  [3XG[103X into the symmetric group of degree [3Xd[103X acting
  transitively on its domain. If [3Xall_ftpr_of_given_degree[103X is set as [3Xtrue[103X, then
  it  returns  a list of all isomorphisms [3XG[103X into the symmetric group of degree
  [3Xd[103X, up to conjugacy equivalence.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xsp := SymplecticGroup(4,2);;[127X[104X
    [4X[25Xgap>[125X [27XFaithfulTransitivePermutationRepresentationsOfDegree(sp,10);[127X[104X
    [4X[28XCompositionMapping( <action epimorphism>, <action isomorphism> )[128X[104X
    [4X[25Xgap>[125X [27XFaithfulTransitivePermutationRepresentationsOfDegree(sp,20, true);[127X[104X
    [4X[28X[ CompositionMapping( <action epimorphism>, <action isomorphism> ), [128X[104X
    [4X[28X  CompositionMapping( <action epimorphism>, <action isomorphism> ), [128X[104X
    [4X[28X  CompositionMapping( <action epimorphism>, <action isomorphism> ) ][128X[104X
  [4X[32X[104X
  
