calculate structure automorphism group in MAGMA

Question

Please hint me! Is there any instruction in MAGMA to calculate structure automorphism group by using generators? for example:

> G:=Graph<16| { 1, 2 }, { 1, 3 }, { 1, 14 }, { 2, 5 }, { 2, 7 }, { 3, 4 }, { 3, 14 },
> { 4, 6 }, { 4, 12 }, { 5, 6 }, { 5, 10 }, { 6, 12 }, { 7, 8 }, { 7, 15 },
> { 8, 9 }, { 8, 15 }, { 9, 10 }, { 9, 11 }, { 10, 11 }, { 11, 13 },
> { 12, 13 }, { 13, 16 }, { 14, 16 }, { 15, 16 }    >;

>  A:=AutomorphismGroup(G);

> Generators(A);

{
    (1, 7)(3, 8)(4, 9)(6, 10)(11, 12)(14, 15),
    (1, 6)(2, 5)(3, 4)(7, 10)(8, 9)(11, 15)(12, 14)(13, 16),
    (1, 11)(2, 13)(3, 9)(4, 8)(5, 16)(6, 15)(7, 12)(10, 14)
}

I want to calculate structure of A? in GAP we have "StructureDescription" to calculate structure of automorphism group:

gap> A:=Group([(1,7)(3,8)(4,9)(6,10)(11,12)(14,15),
> (1,6)(2,5)(3,4)(7,10)(8,9)(11,15)(12,14)(13,16),
> (1,11)(2,13)(3,9)(4,8)(5,16)(6,15)(7,12)(10,14)]);

Group([ (1,7)(3,8)(4,9)(6,10)(11,12)(14,15), 
(1,6)(2,5)(3,4)(7,10)(8,9)(11,15)(12,14)(13,16), 
(1,11)(2,13)(3,9)(4,8)(5,16)(6,15)(7,12)(10,14) ])

gap> StructureDescription(A);

"C2 x C2 x C2"

Answer 1

I'm not exactly sure what you're looking for, but this might be useful.

Since your group $A$ happens to be solvable, the command PCGroup will give you some easily readable information about your group:

PCGroup(A);

GrpPC of order 8 = 2^3
PC-Relations:
    $.1^2 = Id($),
    $.2^2 = Id($),
    $.3^2 = Id($)
Mapping from: GrpPerm: A to GrpPC

It looks like your group is elementary abelian of order 8.

Answer 2

If you have both Magma and GAP at your disposal and if the group in question is not of too large an order, there is a neat little trick I've used a couple of times. Given a group $G$ in Magma, type the following.

IdentifyGroup(G);

This will give you two numbers $a,b$ which identifies the group in the Small Groups Library. Then you can start GAP and type this:

G:=SmallGroup(a,b);
StructureDescription(G);