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We show that an action of SL(2,p), p 7 an odd prime such that 4 6| (p 1), has exactly two orbital digraphs 1 , 2, such that Aut( i) admits a complete block system B of p + 1 blocks of size 2, i =1 , 2, with the following properties: the action of Aut( i) on the blocks of B is nonsolvable, doubly-transitive, but not a symmetric group, and the subgroup of Aut( i )t hat fi xes each block ofB set-wise is semiregular of order 2. If p =2 k 1 > 7 is a Mersenne prime, these digraphs are also Cayley digraphs of the generalized quaternion group of order 2 k+1 .I n this case, these digraphs are non-normal Cayley digraphs of the generalized quaternion group of order 2 k+1 .

Some Non-Normal Cayley Digraphs of the Generalized Quaternion Group of Certain Orders