I am working on a problem where I want to find a presentation of a particular finite group. I have a particular transitive free action I'm interested in, and using this, I've been able to essentially work out the multiplication table for the group. But I would like to come up with a presentation in terms of generators and relations.
After playing around with it, I have found four relations that I believe to be all the relations I need. But I am at a loss for how to prove that these relations are all I need.
I know that formally a presentation is a free group in this case, $\mathcal{F}(x, y)$ quotiented by the normal closure of one of its subgroups. But I don't see this as giving me much leverage on the problem.
The best idea I have right now is to somehow (tediously) work out a procedure for normalizing words into a canonical form.
I am aware that this kind of problem skirts on the word problem and is likely undecidable in general. But what I am looking for is general strategies for attacking these kinds problems.