GAP:ConjugacyClassesSubgroups

Function type
ConjugacyClassesSubgroups is a GAP command that takes as input a group and outputs a list of groups.

Behavior
The command inputs a group and outputs a list of conjugacy classes of subgroups of that group. Each member of the list is thus a list of subgroups, all conjugate to each other and form a full conjugacy class.

Typical use
The typical use of the function is in the form:

ConjugacyClassesSubgroups(group);

(for more, see the section).

Related functions

 * GAP:Subgroups lists all the subgroups of a group.
 * GAP:NormalSubgroups lists only the normal subgroups of the group.
 * GAP:CharacteristicSubgroups lists only the characteristic subgroups of the group.
 * GAP:ConjugateSubgroups takes as input a group and a subgroup and outputs the list of subgroups conjugate to that subgroup in the whole group.

Examples of usage
Here is an example showing the listing of the conjugacy classes of subgroups of a group constructed using the SmallGroup command.

gap> ConjugacyClassesSubgroups(SmallGroup(32,15)); [ Group( of ... )^G, Group( [ f5 ] )^G, Group( [ f3*f4 ] )^G, Group( [ f5, f4 ] )^G, Group( [ f5, f3 ] )^G, Group( [ f5, f3*f4 ] )^G, Group( [ f5, f3, f4 ] )^G, Group( [ f5, f2*f4, f3 ] )^G, Group( [ f5, f1, f4 ] )^G, Group( [ f5, f1*f2, f4 ] )^G, Group( [ f5, f2, f3 ] )^G, Group( [ f5, f3, f4, f1 ] )^G, Group( [ f5, f3, f4, f1*f2 ] )^G, Group( [ f5, f3, f4, f2 ] )^G, Group( [ f5, f3, f4, f1, f2 ] )^G ]

Here is another example that combines the command with the List and Representative commands to output a list of subgroups with one representative of each conjugacy class of subgroups.

gap> List(ConjugacyClassesSubgroups(SymmetricGroup(4)), Representative); [ Group(), Group([ (1,3)(2,4) ]), Group([ (3,4) ]), Group([ (2,4,3) ]), Group([ (1,4)(2,3), (1,3)(2,4) ]), Group([ (1,2)(3,4), (3,4) ]), Group([ (1,2)(3,4), (1,3,2,4) ]), Group([ (3,4), (2,4,3) ]), Group([ (1,3)(2,4), (1,4)(2,3), (1,2) ]), Group([ (1,3)(2,4), (1,4)(2,3), (2,4,3) ]), Group([ (1,3)(2,4), (1,4)(2,3), (2,4,3), (1,2) ]) ]