Question

How do you use GAP to identify the name of a group from its multiplication table? I know that you can define a group from a set of generators, and then look for the group in the set of internal tables:

gap> g := Group([ (1,2), (1,2,3,4,5) ]);    
Group([ (1,2), (1,2,3,4,5) ])

gap> IdGroup(g);                            
[ 120, 34 ]

But how do find out the name of the group [120, 34]?

Answer 1

The StructureDescription command might do what you need. For example,

gap> StructureDescription(g);

returns "S5" for your example, the symmetric group on five elements. For comparison,

gap> StructureDescription(SmallGroup(120,35));

gives "C2 x A5" and

gap> StructureDescription(SmallGroup(120,36));

gives "S3 x (C5 : C4)". I don’t know at what point these descriptions fall below the level of naming the group, but is this on the track you want?