`ConjugacyClassesSubgroups( `

`G` )

`ConjugacyClassesSubgroups`

returns a list of all conjugacy classes of
subgroups of the group `G`. The elements in the list returned are
conjugacy class domains as created by `ConjugacyClassSubgroups`

(see
ConjugacyClassSubgroups). Because conjugacy classes are domains, all
set theoretic functions can be applied to them (see Domains).

In fact, `ConjugacyClassesSubgroups`

computes much more than it returns,
for it calls (indirectly via the function

) the `G`.operations.ConjugacyClassesSubgroups( `G` )`Lattice`

command
(see Lattice), constructs the whole subgroup lattice of `G`, stores it
in the record component

, and finally returns the list
`G`.lattice

. This means, in particular, that it will fail if
`G`.lattice.classes`G` is non-solvable and its maximal perfect subgroup is not in the
built-in catalogue of perfect groups (see the description of the
`Lattice`

command Lattice for details).

gap> # Conjugacy classes of subgroups of S4 gap> s4 := Group( (1,2,3,4), (1,2) );; gap> s4.name := "s4";; gap> cl := ConjugacyClassesSubgroups( s4 ); [ ConjugacyClassSubgroups( s4, Subgroup( s4, [ ] ) ), ConjugacyClassSubgroups( s4, Subgroup( s4, [ (1,2)(3,4) ] ) ), ConjugacyClassSubgroups( s4, Subgroup( s4, [ (3,4) ] ) ), ConjugacyClassSubgroups( s4, Subgroup( s4, [ (2,3,4) ] ) ), ConjugacyClassSubgroups( s4, Subgroup( s4, [ (1,2)(3,4), (1,3)(2,4) ] ) ), ConjugacyClassSubgroups( s4, Subgroup( s4, [ (3,4), (1,2) ] ) ), ConjugacyClassSubgroups( s4, Subgroup( s4, [ (1,2)(3,4), (1,4,2,3) ] ) ), ConjugacyClassSubgroups( s4, Subgroup( s4, [ (2,3,4), (3,4) ] ) ), ConjugacyClassSubgroups( s4, Subgroup( s4, [ (3,4), (1,2), (1,3)(2,4) ] ) ), ConjugacyClassSubgroups( s4, Subgroup( s4, [ (1,2)(3,4), (1,3)(2,4), (2,3,4) ] ) ), ConjugacyClassSubgroups( s4, s4 ) ]

Each entry of the resulting list is a domain. As an example, let us take
the seventh class in the above list of conjugacy classes of *S_4*.

gap> # Conjugacy classes of subgroups of S4 (continued) gap> class7 := cl[7];; gap> # Print the class representative subgroup. gap> rep7 := Representative( class7 ); Subgroup( s4, [ (1,2)(3,4), (1,4,2,3) ] ) gap> # Print the order of the class representative subgroup. gap> Size( rep7 ); 4 gap> # Print the number of conjugates. gap> Size( class7 ); 3

GAP 3.4.4

April 1997