NumberConjugacyClasses( H )
This functions computes the number of conjugacy classes of elements of a group H.
The function uses an algorithm that steps down an elementary abelian
series of the parent group of H and computes the number of conjugacy
classes using the same method as AgGroupOps.ConjugacyClasses does, up
to the last factor group. In the last step the Cauchy-Frobenius-Burnside
lemma is used.
This algorithm is especially designed to supply at least the number of
conjugacy classes of H, whenever ConjugacyClasses fails because of
storage reasons. So one would rather use this function if the last normal
subgroup of the elementary abelian series is too big to be dealt with
ConjugacyClasses.
NumberConjugacyClasses( U, H )
This version of the call to NumberConjugacyClasses computes the number
of conjugacy classes of H under the operation of U. Thus for the
operation to be well defined, U must be a subgroup of the normalizer of
H in their common parent group.
gap> a4 := DerivedSubgroup(s4);;
gap> NumberConjugacyClasses( s4 );
5
gap> NumberConjugacyClasses( a4, s4 );
6
gap> NumberConjugacyClasses( a4 );
4
gap> NumberConjugacyClasses( s4, a4 );
3
GAP 3.4.4