Let's create some group as an example:
gap> G:=Group((1,9,6,7,5)(2,10,3,8,4), (1,10,7,8)(2,9,4,6));
Group([ (1,9,6,7,5)(2,10,3,8,4), (1,10,7,8)(2,9,4,6) ])
It has 8 conjugacy classes of elements:
gap> cc:=ConjugacyClasses(G);
[ ()^G, (3,4,9,7)(5,8,6,10)^G, (3,5,9,6)(4,10,7,8)^G, (3,9)(4,7)(5,6)(8,10)^G,
(2,3,9)(4,10,5)(6,8,7)^G, (1,2)(3,4,5,10,9,7,6,8)^G,
(1,2)(3,7,5,8,9,4,6,10)^G, (1,2,3,7,6)(4,8,5,9,10)^G ]
gap> Length(cc);
8
Let's take the 2nd of them:
gap> c:=cc[2];
(3,4,9,7)(5,8,6,10)^G
It contains 180 elements:
gap> Size(c);
180
and you can test membership of group elements in the conjugacy class as follows:
gap> (3,7,9,4)(5,10,6,8) in c;
true
gap> (3,7,6,8) in c;
false
Conjugacy class is not internally represented as a list of its elements - that would be very inefficient (for example, for algorithms that need only representatives of conjugacy classes). But if you need to get a list if all elements of the class, you can get them as follows:
gap> AsList(c);
[ (3,4,9,7)(5,8,6,10), (3,7,9,4)(5,10,6,8), (2,6,9,10)(4,7,8,5),
(2,10,9,6)(4,5,8,7), (2,10,7,5)(3,6,8,9), (2,5,7,10)(3,9,8,6),
(2,3,6,7)(4,9,10,8), (2,7,6,3)(4,8,10,9), (2,9,5,4)(3,8,10,7),
...
(1,9,6,10)(3,7,4,8), (1,10,6,9)(3,8,4,7), (1,3,4,10)(5,6,8,9),
(1,10,4,3)(5,9,8,6), (3,8,9,10)(4,5,7,6), (3,10,9,8)(4,6,7,5) ]
This may be very memory inefficient for large groups, but you can also iterate over its elements as follows:
gap> for x in c do
> Print(x,"\n");
> od;
( 3, 4, 9, 7)( 5, 8, 6,10)
( 3, 7, 9, 4)( 5,10, 6, 8)
...
( 3, 8, 9,10)( 4, 5, 7, 6)
( 3,10, 9, 8)( 4, 6, 7, 5)
without constructing the whole list, and also use enumerator which will give you a list-like behaviour:
gap> enum:=Enumerator(c);
<enumerator of (3,4,9,7)(5,8,6,10)^G>
gap> enum[2];
(3,7,9,4)(5,10,6,8)
gap> Position( enum, (3,7,9,4)(5,10,6,8) );
2
gap> Position( enum, (3,7,6,8) );
fail
also without constructing the whole list.
Best Answer
First, you may find helpful some functions like
NormalSubgroups,IsomorphicSubgroupsetc. which are listed in "How do I get the subgroups of my group?" entry from the GAP F.A.Q. There are also further functions not listed there, such asRepresentativesPerfectSubgroups,RepresentativesSimpleSubgroups,ConjugacyClassesMaximalSubgroups- see Chapter "Groups" from the GAP manual for more functions with the similar semantics.Regarding p-subgroups, the manual entry for
IsomorphicSubgroupssuggests the same approach as in the Derek Holt's comment above: "To find p-subgroups it is often faster to compute the subgroup lattice of the Sylow subgroup".Furthermore,
LatticeByCyclicExtensionandSubgroupsSolvableGroupaccept optional arguments which allow to put restrictions on computed subgroups. In the latter case functionsSizeConsiderFunctionandExactSizeConsiderFunctionmay be used. All four functions mentioned in this paragraph are documented in the Chapter "Groups", type e.g.?SubgroupsSolvableGroupto see their descriptions in GAP.As for tables of marks mentioned by Jack Schmidt, see Chapter "Tables of Marks" and in particular Section "Accessing Subgroups via Tables of Marks" - but that will work only for groups whose tables are contained in the tables of marks library.