University of Isfahan International Journal of Group Theory 2251-7650 5 3 2016 09 01 Normal edge-transitive and ½ -ARC-transitive cayley graphs on non-abelian groups of order 2pq, p > q are odd primes 1 8 6537 10.22108/ijgt.2016.6537 EN Ali Reza Ashrafi University of Kashan Bijan Soleimani University of Kashan Journal Article 2014 01 16 Darafsheh and Assari in [Normal edge-transitive Cayley graphs on non-abelian groups of order $4p$, where $p$ is a prime number, <em>Sci. China Math.</em>, <strong>56</strong> (2013) 213--219.] classified the connected normal edge transitive and $\frac{1}{2}-$arc-transitive Cayley graph of groups of order $4p$. In this paper we continue this work by classifying the connected Cayley graph of groups of order $2pq$, $p > q$ are primes. As a consequence it is proved that $Cay(G,S)$ is a $\frac{1}{2}-$arc-transitive Cayley graph of order $2pq$, $p > q$ if and only if $|S|$ is an even integer greater than 2, $S = T \cup T^{-1}$ and $T \subseteq \{ cb^ja^{i} \ | \ 0 \leq i \leq p - 1\}$, $1 \leq j \leq q-1$, such that $T$ and $T^{-1}$ are orbits of $Aut(G,S)$ and<br /><br />$G ≅ < a, b, c  |  a^p = b^q = c^2 = e, ac = ca, bc = cb, b^{-1}ab = a^r >$,  or<br /><br />$G ≅ < a, b, c  |  a^p = b^q = c^2 = e, c ac = a^{-1}, bc = cb, b^{-1}ab = a^r >$,<br /><br />where $r^q \equiv 1  (mod p)$.<br />  https://ijgt.ui.ac.ir/article_6537_5d2a53752a30743d1750e751249611aa.pdf