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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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The quaternion and dihedral groups of order $8$. – Torsten Ekedahl May 29 '10 at 17:38
up vote 13 down vote accepted

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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This is for $n\ge5$. – Gerry Myerson May 30 '10 at 4:58

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