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I have three groups here: <a,b,c,d | [ab,cd]=1>; <a,b,c,d | abcda^(-1)b^(-1)c^(-1)d^(-1)=1>; <a,b,c,d | [a,b][c,d]=1>.

I finally got three Cayley graphs down, first one is like Z^2, but with each vertice has eight other vertices come out due to the fact of eight generators. Second and third Cayley graphs both have to be done on hyperbolic plane as it is octagons with each vertice has other eight octagons. I have checked the graphs are right.

I thought to try to least find automorphism for simple Z^2, which would just have cayley graphs as grid lines, but I'm even quite confused with how to do this. By starting looking at some symmetries, what should I do next then?

Also, how can I find automorphisms of above three Cayley graphs? Can anyone please help me a bit? Thanks a lot.