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Motivating factor behind this question is the comments which are given for this question asked some hours ago http://math.stackexchange.com/questions/6538/group-of-order-105

So given a group $G$ of order $n$ what are the different methods for constructing a non abelian group of order $n$. Well, i have seen a method which Herstein uses in his book. He takes a cyclic group of given order and defines an Automorphism on the cyclic group and then places some restrictions.

I would like to know, whether there are anymore methods for obtaining a Non Abelian group of a given order.

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See mathoverflow.net/questions/31553/… –  Robin Chapman Oct 11 '10 at 18:37
Also mathoverflow.net/questions/11001/…. –  Arturo Magidin Oct 11 '10 at 21:09
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up vote 2 down vote accepted

Have you tried taking the semi-direct product of two subgroups (one needs to be a normal subgroup). The order of G is the product of the orders of the subgroups. One subgroup needs to be normal. If the other is not, then the semidirect product is usually (if not always) non-abelian

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Well, yes, but to get a non-Abelian semidirect product, you need to know that there is some non-trivial homomorphism from that group into the automorphism group of the normal one, which is the hard part- there's always a homomorphism with kernel the whole group, which leads to the direct product. –  Geoff Robinson Jul 8 '11 at 10:44
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