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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.

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    $\begingroup$ The quaternion and dihedral groups of order $8$. $\endgroup$
    – Torsten Ekedahl
    Commented May 29, 2010 at 17:38

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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).

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    $\begingroup$ This is for $n\ge5$. $\endgroup$
    – Gerry Myerson
    Commented May 30, 2010 at 4:58

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