I know this isn't exactly what was asked, but a few weeks ago at CGT,
Derek Holt spoke about finding certain kinds of subgroups of S_n up to
conjugacy. Up to 4095, the primitive ones are known. Up to 32 the
transitive ones are known. And up to 18 ALL of the (conjugacy classes
of) subgroups of S_n are known.
There's a paper about finding the transitive ones on his website (which
references a paper of Alexander Hulpke's on the same subject).
Joe
Thomas Breuer wrote:
proposed answer to the GAP Forum
------------------------------------------------------------------------
Dear GAP Forum,
Max Horn asked:
is there an "efficient" way to compute (the conjugacy classes of) all
transitive subgroups of SymmetricGroup(n) or order n?
I.e. something more efficient than
Filtered(ConjugacyClassesSubgroups(SymmetricGroup(n)),
x -> n=Size(Representative(x)) and IsTransitive(Representative(x)))
which has to compute all conjugacy classes.
Any group of order n has only one transitive permutation representation
on n points, up to equivalence: its regular permutation representation.
That is, the classes of transitive groups of degree and order n
are given by the classes of regular permutation representations of
the groups of order n.
So if one knows the groups of order n, up to isomorphism,
then one can write down the required subgroups of the symmetric group
of degree n.
(This condition is satisfied for not too large n,
via the GAP library of small groups;
see "Small Groups" in the GAP Reference Manual.)
Here is one possibility to do this explicitly in GAP.
List( AllSmallGroups( n ), G -> Action( G, Elements( G ), OnRight ) );
All the best,
Thomas
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