# Groups of order $2pq$ where $p,q$ are odd primes, $p<q$ and $q+1\neq 2p$

Suppose that $G$ is a finite group of order $2pq$, where $p,q$ are odd primes, $p<q$ and $q+1\neq 2p$.

We know by Sylow's theorem that $G$ has only one Sylow $q$-subgroup (say $S$); so $S \unlhd G$, and $|G/S|=2p$. Therefore $G/S\cong C_{2p}$ or $D_{2p}$.

Can we classify all such finite groups $G$?

The group has order 2pq, where $2<p<q$
Sure, in fact, it's easy to classify all groups of order $2pq$, where $p$ and $q$ are primes and $2<p<q$.
There's always the following four: $C_{2pq}$, $D_{2pq}$, $C_p\times D_{2q}$ and $C_q\times D_{2p}$. Moreover, if $p$ divides $q-1$, then we will get an extra two: $(C_q\rtimes C_p)\times C_2$ and $C_q\rtimes C_{2p}$.
• Please, stop changing your question (s). The short answer is that groups of order $2pq$ are easy to classify, and most of the questions you have about them follow easily from the classification. If you have trouble understanding the list of groups given, or would like to know more about how to prove this result, then say so. – verret Mar 23 '16 at 7:30