What are the conjugacy classes of the group $\{k,r \,| \, k^3=r^2=(kr)^4=1\}$?

Question

How can I determine the conjugacy classes of a group if I have the presentation of the group? For example, we know that $S_4$ has the presentation $$ S_4=\{k,r \,| \, k^3=r^2=(kr)^4=1\}. $$ What are the conjugacy classes of it?

Answer 1

I assume you mean the conjugacy classes of elements. This may be an exhaustive way, but with the given relations, you can calculate all of the elements of your group, e.g.

$k,k^2,r,kr,krkr,krkrkr$

then any combinations of those

$k^2r, k^2rkr, k^2rkrkr, rk, rk^2, rkr,...$ and so on. Since you have more relators than generators, this step ends after finite steps.

Finally you can calculate all the conjugate elements. Use your relations to eliminate duplicates.

Unfortunately this is a very inefficient way. I would rather suggest to get a permutation representation. There, you can write down every conjugacy class immediately, as the permutations of the same cycle types belong to the same class.

You could also check some books about computational group theory. Probably they will offer a better algorithm.