In this paper we determine all tetravalent Cayley graphs of a non-abelian group of order 3p2,where p is a prime number greater than 3,and with a cyclic Sylow p-subgroup.We show that all of these tetravalent Cayley graphs are normal.The full automorphism group of these Cayley graphs is given and the half-transitivity and the arc-transitivity of these graphs are investigated.We show that this group is a 5-CI-group.