These three commands form an interface to the transitive groups database in the GroupTheory package.  Currently, conjugacy class representatives of the transitive groups up to degree $37$, but excluding those of degree $32$, are available in the database.

The NumTransitiveGroups( d ) command returns the number of transitive groups of degree d stored in the transitive groups database.  If the value returned is $0$, this indicates that no transitive groups of degree d are in the database.  (It does not indicate that there are no transitive groups of that degree!)

The TransitiveGroup( d, k ) command returns the $k$-th transitive group of degree $d$ from the transitive groups database.  If $k$ is larger than the number of transitive groups of degree d, an exception is raised.

The AllTransitiveGroups( d ) command returns a list of all the transitive groups of degree $d$.  It is essentially equivalent to [seq]( TransitiveGroup( d, k ), k = 1 .. NumTransitiveGroups( d ) ), but avoids some repeated checks.

$\mathrm{with}\left(\mathrm{GroupTheory}\right)\:$

$\mathrm{NumTransitiveGroups}\left(31\right)$

$12$

$G≔\mathrm{TransitiveGroup}\left(31\,3\right)$

$G≔⟨\left(2\,12\,22\right)\left(3\,13\,23\right)\left(4\,14\,24\right)\left(5\,15\,25\right)\left(6\,16\,26\right)\left(7\,17\,27\right)\left(8\,18\,28\right)\left(9\,19\,29\right)\left(10\,20\,30\right)\left(11\,21\,31\right)\,\left(1\,2\,26\,3\,20\,22\,27\,30\,14\,4\,16\,25\,21\,13\,24\,23\,8\,9\,28\,6\,10\,31\,19\,29\,15\,12\,7\,5\,18\,11\,17\right)⟩$

$\mathrm{IsTransitive}\left(G\right)$

$\mathrm{true}$

$\mathrm{IsPrimitive}\left(G\right)$

$\mathrm{true}$

$L≔\mathrm{AllTransitiveGroups}\left(31\right)\:$$\mathrm{nops}\left(L\right)$

$12$

$\mathrm{andmap}\left(\mathrm{IsTransitive}\,L\right)$

$\mathrm{true}$

$\mathrm{andmap}\left(\mathrm{IsPrimitive}\,L\right)$

$\mathrm{true}$

$\mathrm{NumTransitiveGroups}\left(8\right)$

$50$

$\mathrm{nops}\left(\mathrm{remove}\left(\mathrm{IsPrimitive}\,\mathrm{AllTransitiveGroups}\left(8\right)\right)\right)$

$43$

The GroupTheory[TransitiveGroup], GroupTheory[NumTransitiveGroups] and GroupTheory[AllTransitiveGroups] commands were introduced in Maple 17.

For more information on Maple 17 changes, see Updates in Maple 17.