Given a small number of generators for a group of small (e.g. $3\times3$) matrices over a finite field $\mathbb{F}_p$ where $p$ is prime, how can we find all the elements of the group?
The best I've come up with so far is to keep taking all possible products of two elements of the group until we no longer get any new elements. This is fine when the groups are small, and it's the algorithm used here: . But some of the groups I'm considering have order over 10000 and therefore need over 100 million matrix multiplications and comparisons, which is taking Matlab far too long to compute!
Is there a way to know when we've got the whole group which is quicker than pairwise multiplication?
Many thanks for any help with this!