# Software or tool for investigating groups

I'm interested in software that has the ability to investigate finite groups. In particular, I'd like to be able to ask it questions like "What are the solutions to $x^3 = 1$?" (i.e. find cube roots of unity) for a permutation group.

I've had a look at Maxima but don't see a way to ask it do that.

• You can say, for example, Filtered(SymmetricGroup(6), x -> x*x*x = ()); to get the list of elements of order three. I'm sure there are smarter ways :-) Aug 29, 2014 at 19:29
• In larger groups, you could start by finding conjugacy class representatives and then looking for elements of order $3$ among these. Or if the order of the group is divisible by a relatively small power of $3$, you could first find a Sylow $3$-subgroup and look for its conjugacy classes. Aug 30, 2014 at 16:25