  
  [1X2 [33X[0;0YObtaining Core-Free Subgroups[133X[101X
  
  
  [1X2.1 [33X[0;0YCore-Free Subgroups[133X[101X
  
  [33X[0;0YA core-free subgroup is a subgroup in which its (normal) core is trivial.[133X
  
  [1X2.1-1 IsCoreFree[101X
  
  [33X[1;0Y[29X[2XIsCoreFree[102X( [3XG[103X, [3XH[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10Ya boolean[133X
  
  [33X[0;0YGiven  a  group  [3XG[103X  and  one  of  its  subgroups  [3XH[103X, it returns whether [3XH[103X is
  core-free in [3XG[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XG := SymmetricGroup(4);; H := Subgroup(G, [(1,3)(2,4)]);;[127X[104X
    [4X[25Xgap>[125X [27XCore(G,H);[127X[104X
    [4X[28XGroup(())[128X[104X
    [4X[25Xgap>[125X [27XIsCoreFree(G,H);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XH := Subgroup(G, [(1,4)(2,3), (1,3)(2,4)]);;[127X[104X
    [4X[25Xgap>[125X [27XIsCoreFree(G,H);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XCore(G,H);# H is a normal subgroup of G, hence it does not have a trivial core[127X[104X
    [4X[28XGroup([ (1,4)(2,3), (1,3)(2,4) ])[128X[104X
  [4X[32X[104X
  
  [1X2.1-2 CoreFreeConjugacyClassesSubgroups[101X
  
  [33X[1;0Y[29X[2XCoreFreeConjugacyClassesSubgroups[102X( [3XG[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10Ya list[133X
  
  [33X[0;0YReturns a list of all conjugacy classes of core-free subgroups of [3XG[103X[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XG := SymmetricGroup(4);; dh := DihedralGroup(10);;[127X[104X
    [4X[25Xgap>[125X [27Xcfccs := CoreFreeConjugacyClassesSubgroups(G);; Size(cfccs);[127X[104X
    [4X[28X7[128X[104X
    [4X[25Xgap>[125X [27Xcfccs_dh := CoreFreeConjugacyClassesSubgroups(dh);; Size(cfccs_dh);[127X[104X
    [4X[28X2[128X[104X
  [4X[32X[104X
  
  [1X2.1-3 AllCoreFreeSubgroups[101X
  
  [33X[1;0Y[29X[2XAllCoreFreeSubgroups[102X( [3XG[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10Ya list[133X
  
  [33X[0;0YReturns a list of all core-free subgroups of [3XG[103X[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XG := SymmetricGroup(4);; dh := DihedralGroup(10);;[127X[104X
    [4X[25Xgap>[125X [27Xacfs := AllCoreFreeSubgroups(G);; Size(acfs);[127X[104X
    [4X[28X24[128X[104X
    [4X[25Xgap>[125X [27Xacfs_dh := AllCoreFreeSubgroups(dh);; Size(acfs_dh);[127X[104X
    [4X[28X6[128X[104X
  [4X[32X[104X
  
  
  [1X2.2 [33X[0;0YDegrees of Core-Free subgroups[133X[101X
  
  [1X2.2-1 CoreFreeDegrees[101X
  
  [33X[1;0Y[29X[2XCoreFreeDegrees[102X( [3XG[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10Ya list[133X
  
  [33X[0;0YReturns  a  list  of all possible degrees of faithful transitive permutation
  representations  of  [3XG[103X.  The  degrees  of  a faithful transitive permutation
  representation of [3XG[103X are the index of its core-free subgroups.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XG := SymmetricGroup(4);; dh := DihedralGroup(10);;[127X[104X
    [4X[25Xgap>[125X [27XCoreFreeDegrees(G);[127X[104X
    [4X[28X[ 4, 6, 8, 12, 24 ][128X[104X
    [4X[25Xgap>[125X [27XCoreFreeDegrees(dh);[127X[104X
    [4X[28X[ 5, 10 ][128X[104X
  [4X[32X[104X
  
