Getting the multiplication table in Magma

Question

I have described a group in Magma by specifying a presentation:

F<x000_0,x000_1,x000_2,x001_0,x001_1,x001_2,x010_0,x010_1,x010_2,x011_0,x011_1,x011_2,x100_0,x100_1,x100_2,x101_0,x101_1,x101_2,x110_0,x110_1,x110_2,x111_0,x111_1,x111_2> := FreeGroup(24);
G<x000_0,x000_1,x000_2,x001_0,x001_1,x001_2,x010_0,x010_1,x010_2,x011_0,x011_1,x011_2,x100_0,x100_1,x100_2,x101_0,x101_1,x101_2,x110_0,x110_1,x110_2,x111_0,x111_1,x111_2>, phi := quo<F | x000_0,x000_0=x100_0,x000_0 * x000_1 = x000_1,x000_0 * x000_2 = x000_2,x000_1=x010_1,x000_1 * x000_1 = x000_2,x000_1 * x000_2 = x000_0,x000_2=x001_2,x000_2 * x000_1 = x000_0,x000_2 * x000_2 = x000_1,x001_0=x101_0,x001_0 * x100_1 = x001_1,x001_0 * x100_2 = x001_2,x001_1=x011_1,x001_1 * x100_1 = x001_2,x001_1 * x100_2 = x001_0,x001_2=x000_2,x001_2 * x100_1 = x001_0,x001_2 * x100_2 = x001_1,x010_0=x110_0,x010_0 * x100_1 = x010_1,x010_0 * x100_2 = x010_2,x010_1=x000_1,x010_1 * x100_1 = x010_2,x010_1 * x100_2 = x010_0,x010_2=x011_2,x010_2 * x100_1 = x010_0,x010_2 * x100_2 = x010_1,x011_0=x111_0,x011_0 * x000_1 = x011_1,x011_0 * x000_2 = x011_2,x011_1=x001_1,x011_1 * x000_1 = x011_2,x011_1 * x000_2 = x011_0,x011_2=x010_2,x011_2 * x000_1 = x011_0,x011_2 * x000_2 = x011_1,x100_0=x000_0,x100_0 * x100_1 = x100_1,x100_0 * x100_2 = x100_2,x100_1=x110_1,x100_1 * x100_1 = x100_2,x100_1 * x100_2 = x100_0,x100_2=x101_2,x100_2 * x100_1 = x100_0,x100_2 * x100_2 = x100_1,x101_0=x001_0,x101_0 * x000_1 = x101_1,x101_0 * x000_2 = x101_2,x101_1=x111_1,x101_1 * x000_1 = x101_2,x101_1 * x000_2 = x101_0,x101_2=x100_2,x101_2 * x000_1 = x101_0,x101_2 * x000_2 = x101_1,x110_0=x010_0,x110_0 * x000_1 = x110_1,x110_0 * x000_2 = x110_2,x110_1=x100_1,x110_1 * x000_1 = x110_2,x110_1 * x000_2 = x110_0,x110_2=x111_2,x110_2 * x000_1 = x110_0,x110_2 * x000_2 = x110_1,x111_0=x011_0,x111_0 * x100_1 = x111_1,x111_0 * x100_2 = x111_2,x111_1=x101_1,x111_1 * x100_1 = x111_2,x111_1 * x100_2 = x111_0,x111_2=x110_2,x111_2 * x100_1 = x111_0,x111_2 * x100_2 = x111_1>;

I am getting a group of order $12$. The function IdentifyGroup recognizes it as <12,3>. I hope this means $A_4$.

Anyway, now I would like to know the product of each pair of generators. In this case, if I am not mistaken, the product of any two generators is a generator itself.

Therefore, it would suffice to be able to check, for each triple of generators $a,b,c$, whether or not $ab=c$. How can this be done?

Answer 1

To do what you ask, you could turn the group into a rewriting system. But note that you have more generators than group elements, so some of them will be equal.

> R:=RWSGroup(G);
> R.1*R.11;
R.5
> R.23;
R.9^-1
> Seq(R,0,1);
[ Id(R), R.2, R.2^-1, R.4, R.5, R.5^-1, R.7, R.9, R.9^-1, R.10, R.14, R.14^-1 ]

For other types of calculations it might be more useful to turn it into a permutation group. The geenrators of $P$ will still correspond to those of $G$.

> P:=CosetImage(G,sub<G|>);
> IsIsomorphic(P,Alt(4));
true.
> P.23 eq P.9^-1;
true