[GAP Forum] Transitive subgrops of Sym(n) of order n

[Thomas Breuer]

proposed answer to the GAP Forum
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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