Normal edge-transitive and 1/2-arc-transitive semi-Cayley graphs

Darafsheh and Assari in [Normal edge-transitive Cayley graphs on non-abelian groups of order 4p‎, ‎where p is a prime number‎, ‎Sci‎. ‎China Math‎., ‎56 (1) (2013) 213-219.] classified the connected normal edge transitive and‎ ‎12−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 ‎12−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 ‎\begin{eqnarray*}‎ ‎G \cong \langle a‎, ‎b‎, ‎c \ | \ a^p = b^q = c^2 = e‎, ‎ac = ca‎, ‎bc = cb‎, ‎b^{-1}ab = a^r \rangle‎, ‎\ or\\‎ ‎G \cong \langle a‎, ‎b‎, ‎c \ | \ a^p = b^q = c^2 = e‎, ‎c ac = a^{-1}‎, ‎bc = cb‎, ‎b^{-1}ab = a^r \rangle‎, ‎\end{eqnarray*}‎ ‎where r^q \equiv 1 \ (mod p)‎.