$G$ is defined by the relations $x^m=y^n=1,xy=yx^k$. For which $m,n,k$ does this give a nonabelian group order $mn$?
I started playing around with this using GAP, starting with the case where $m=13$.
For $n=3$ only $k=3,9$ worked, and they generated isomorphic groups, similarly for $n=9$. For $n=5,7,11$ no value of $k$ worked. For $n=2$, $k=12$ worked. For $n=4$, $k=5,8$ worked and generated isomorphic groups. $k=12$ also worked but generated a non-isomorphic group. Similarly for $n=8$. For $n=6$, $k=3,4,9,10,12$ worked. The groups for $k=3,9$ were isomorphic. and those for $k=4,10$ were isomorphic.
For $n=12$, $k=2,\dots,12$ all worked. Those for $k=2,6,7,11$ were isomorphic, those for $k=3,9$ were isomorphic, those for $k=4,10$ were isomorphic, and those for $k=5,8$ were isomorphic.
I am having difficulty figuring out what is going on. Is all this well-known material? Is there a complete or partial answer to the opening question (for which $m,n,k$ do we get a non-abelian group order $mn$)? If so, where can I find it?
Also how does one do this stuff by hand? I tried a few cases and found it quite tricky to prove that $x=1$, as often happened for the $k$ that failed.