Let G and H be finite groups. Let G' = [G,G] and H' = [H,H] be the corresponding derived groups (commutator subgroups) of G and H. I am looking for an example where G' is isomorphic to H' and G/G' is isomorphic to H/H' but G and H are not isomorphic.



Must finite groups with isomorphic commutators and quotients be isomorphic?
take $G=S\_n$ and $H=A\_n\times Z/2Z$ (here $S\_n$ is the symmetric group and $A\_n$ is the alternating group).


