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GitHub Repository: sagemathinc/cocalc-example-files
 Path: blob/master/gap/gap.sagews
Views: 968
gap.version()
'4.5.7' 

A = AbelianGroup(3, [2,3,4], names="abc")
A
Multiplicative Abelian group isomorphic to C2 x C3 x C4 
a,b,c = A.gens()
(c^3*b).list()
[0, 1, 3] 
A.permutation_group()
Permutation Group with generators [(6,7,8,9), (3,4,5), (1,2)] 
S5 = SymmetricGroup(5)
A5 = AlternatingGroup(5)
A5.is_subgroup(S5)
True 
A5.gens()
[(3,4,5), (1,2,3,4,5)] 
S5_gap = gap(S5)
A5_gap = gap(A5)

S5_gap.DerivedSeries()
[ SymmetricGroup( [ 1 .. 5 ] ), AlternatingGroup( [ 1 .. 5 ] ) ] 
A5_gap.CompositionSeries()
[ AlternatingGroup( [ 1 .. 5 ] ), Group( () ) ] 
for cjcls in A5_gap.ConjugacyClasses():
    print cjcls
    print cjcls.Orbits()
    print
ConjugacyClass( AlternatingGroup( [ 1 .. 5 ] ), () ) [ [ () ] ] ConjugacyClass( AlternatingGroup( [ 1 .. 5 ] ), (1,2)(3,4) ) [ [ (1,2)(3,4), (2,3)(4,5), (1,2)(4,5), (1,5)(3,4), (2,4)(3,5), (1,5)(2,3), (1,2)(3,5), (1,3)(4,5), (1,4)(3,5), (2,5)(3,4), (1,3)(2,4), (1,4)(2,3), (1,5)(2,4), (1,4)(2,5), (1,3)(2,5) ] ] ConjugacyClass( AlternatingGroup( [ 1 .. 5 ] ), (1,2,3) ) [ [ (1,2,3), (2,3,4), (1,2,4), (3,4,5), (2,4,5), (2,3,5), (1,2,5), (1,4,5), (1,3,5), (2,5,3), (1,3,4), (2,4,3), (1,5,3), (1,4,3), (3,5,4), (2,5,4), (1,4,2), (1,5,4), (1,5,2), (1,3,2) ] ] ConjugacyClass( AlternatingGroup( [ 1 .. 5 ] ), (1,2,3,4,5) ) [ [ (1,2,3,4,5), (1,2,4,5,3), (1,4,2,3,5), (1,2,5,3,4), (1,5,2,4,3), (1,4,5,2,3), (1,3,5,4,2), (1,3,2,5,4), (1,3,4,2,5), (1,5,3,2,4), (1,4,3,5,2), (1,5,4,3,2) ] ] ConjugacyClass( AlternatingGroup( [ 1 .. 5 ] ), (1,2,3,5,4) ) [ [ (1,2,3,5,4), (1,5,2,3,4), (1,2,4,3,5), (1,3,4,5,2), (1,3,2,4,5), (1,2,5,4,3), (1,4,5,3,2), (1,4,2,5,3), (1,5,4,2,3), (1,4,3,2,5), (1,5,3,4,2), (1,3,5,2,4) ] ]