Using GAP to Define a Group

Question

How do you use GAP to determine the group name from the multiplication table? I know that you can define a group from a set of generators and then look for a group in a 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 you know the name of the group [120, 34]?

Answer 1

The StructureDescription command (see section 36.7 of the GAP manual ) can do what you need. For instance,

StructureDescription(g)

returns "S5" for your example, telling you that (as you probably already know) this is a symmetric group of five elements.

For comparison,

StructureDescription(SmallGroup(120,35))

gives "C2 x A5" and

StructureDescription(SmallGroup(120,36))

"S3 x (C5: C4)". , , ?