If $p < q$ are primes then there is a nonabelian group of order $pq$ iff $q = 1 \pmod p$, 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"?



Generalizations of the nonabelian group of order $pq$
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 $a^p=1$, $b^n=1$, $a^{-1}ba=b^r$. 


