Question

If p < q are primes then there is a nonabelian group of order pq iff q = 1 mod p, in which case the group is unique. If p = 2 we obtain the dihedral group of order 2q, which generalizes first to the dihedral group of order 2n and then even further to the "generalized dihedral group" where the cyclic group of order n is replaced with any abelian group.

What if p > 2? Is there a natural generalization of the groups of order pq to a family of groups of order pn? Maybe more than one possible generalization? Is it maybe even meaningful to talk about the "generalized p-hedral group"?

Answer 1

The nonabelian group of order $pq$ is given by generators $a$, $b$, with relations $a^p=1$, $b^q=1$, $a^{-1}ba=b^r$, where $r$ is chosen so $r^p$ is 1 mod $q$. If there is an element $r$ of order $p$ mod $n$, then there is a nonabelian group of order $pn$ with generators $a$, $b$, and relations $a^p=1$, $b^n=1$, $a^{-1}ba=b^r$.