Normal edge-transitive and $frac{1}{2}-$arc$-$transitive Cayley graphs on non-abelian groups of order $2pq$, $p > q$ are odd primes

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.] classied the connected normal edge transitive and 1 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 1 edgetransitive Cayley graph of order 2pq, p > q if and only if jSj is an even integer greater than 2, S = T[ T 1 and T f cba i j 0 i p 1g such that T and T 1 are orbits of Aut(G;S) and