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Dear GAP Forum,

Igor Schein asked

I have a somewhat related question. There're 51 groups of order 32, and

some of them are very hard to distinguish. For example,

SmallGroup(32,27) and SmallGroup(32,34). Are there standard methods (

preferable ones that can be done in GAP ) to determine whether or not 2

groups like above are isomorphic?

For small groups, the function `IdGroup' can be used to distinguish

isomorphism types.

gap> g1:= SmallGroup(32,27);; gap> IdGroup( g1 ); [ 32, 27 ] gap> g2:= SmallGroup(32,34);; gap> IdGroup( g2 ); [ 32, 34 ]

This works of course not only for groups fetched from the library.

gap> IdGroup( SymmetricGroup( 4 ) ); [ 24, 12 ]

If one also needs the isomorphism or if the groups are too big for

being handled by `IdGroup' then one can use `IsomorphismGroups'.

This function returns an isomorphism if the two groups are isomorphic,

and `fail' otherwise.

gap> IsomorphismGroups( g1, g2 ); fail gap> IsomorphismGroups( SymmetricGroup( 4 ), SmallGroup( 24, 12 ) ); [ (1,2,3,4), (1,2) ] -> [ f1*f2*f3*f4, f1*f3*f4 ]

(See also the GAP manual for more details.)

Kind regards,

Thomas

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