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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 $G$?

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The group has order 2pq, where $2<p<q$

Hint: The possibilities are that G is either a direct product of cyclic groups (which is only one case) or some kinds of semi-direct products of cyclic groups.

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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 $C_q\rtimes C_{2p}$.

(There's various ways to prove the above. One way which is a bit heavy handed, is to appeal to the classification of groups of squarefree order.)

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  • $\begingroup$ Have you finished changing your question? $\endgroup$ – verret Mar 23 '16 at 7:23
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    $\begingroup$ 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. $\endgroup$ – verret Mar 23 '16 at 7:30

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