There is a systematic mechanism when classifying groups of order pq, p and q primes with p < q and p | q-1.
Below is to find a nonabelian group of order 21 using a semidirect product.
Let G be group of order 21. Let P be a Sylow-3 subgroup of G, , and let . If (q-1)=6 were not divisible by p=3. It is easy. P and Q are unique sylow subgroups in G and . However, (q-1)=6 is divisible by p=3.
In this case, we use a semidirect product and , for some (We know that Q is a unique sylow-7 subgroup in G, thus it is normal in G. However, P is not a unique sylow-3 subgroup in G).
Since and is divisible by , where x in , we see that is not trivial.
Let , ; let . Then with some relationships, where and
Consider . We see that is defined by here since is cyclic, containing a unique subgroup of order 3 by Cauchy's theorem; the latter parenthesis is used to denote the automorphism of order 3.
Thus we have , where .