# Finite groups of the form p^3q

Hi,

Is there any way to classify finite non-abelian groups of order p^3q with p and q primes?

Thanks.

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Yes, either a Sylow p-group is normal or a Sylow q-group is normal, unless your group has order 24. –  Steve D May 3 '11 at 22:45
And even in the order 24 case, $S_4$ is the only exception. –  Steve D May 3 '11 at 22:46
This is indeed a well known fact. In case you want a reference, I can strongly recommend Martin Isaac's book "Finite Group Theory". The theorem that Steve mentioned is in Theorems 1.32 and 1.33 of that book. –  Tom De Medts May 4 '11 at 7:18

If you would like a historical reference, Burnside in his book cites Western, "Groups of order $p^3q$", Proc. LMS Vol xxx. 1899 pp. 209--263, But the length of that paper suggests another source would be better if all you want is the result...