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Hello,

You made a mistake. The database of AllGroups is about all groups up to isomorphisms. Some of them are given by their presentation, others as permutation groups... but there is no canonical way to convert all of them to a permutation group.

Note that there is also a database of transitive permutation groups that can be used as follows

sage: G = gap.TransitiveGroup(12,5)
sage: PermutationGroup(gap_group = G)
Permutation Group with generators [(1,5,9)(2,6,10)(3,7,11)(4,8,12), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12), (1,8,7,2)(3,6,9,12)(4,11,10,5)]

And this is actually directly available in Sage with

sage: 
sage: G = TransitiveGroup(12,4)
sage: G
Transitive group number 4 of degree 12
sage: G.gens()
[(1,9,5)(2,4,3)(6,8,7)(10,12,11), (1,11,6)(2,9,7)(3,10,5)(4,8,12)]

Vincent

Hello,

Note that there is also a database of transitive permutation groups that can be used as follows

sage: G = gap.TransitiveGroup(12,5)
sage: PermutationGroup(gap_group = G)
Permutation Group with generators [(1,5,9)(2,6,10)(3,7,11)(4,8,12), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12), (1,8,7,2)(3,6,9,12)(4,11,10,5)]

And this is actually directly available in Sage with

sage: 
sage: G = TransitiveGroup(12,4)
sage: G
Transitive group number 4 of degree 12
sage: G.gens()
[(1,9,5)(2,4,3)(6,8,7)(10,12,11), (1,11,6)(2,9,7)(3,10,5)(4,8,12)]

EDIT: there is also the possibility to look at the list of groups of given size up to isomorphisms but I think that it can be done only through the gap interface (or in gap directly)

sage: groups = gap.AllGroups(12)
sage: for g in groups:
....:     print g
....:     print gap.ConjugacyClasses(g)
....:     
Group( [ f1, f2, f3 ] )
[ ConjugacyClass( Group( [ f1, f2, f3 ] ), <identity> of ... ), 
  ConjugacyClass( Group( [ f1, f2, f3 ] ), f1 ), 
  ConjugacyClass( Group( [ f1, f2, f3 ] ), f2 ), 
  ConjugacyClass( Group( [ f1, f2, f3 ] ), f3 ), 
  ConjugacyClass( Group( [ f1, f2, f3 ] ), f1*f2 ), 
  ConjugacyClass( Group( [ f1, f2, f3 ] ), f2*f3 ) ]
...

Vincent