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

#### Abstract

We show that an action of SL$(2,p)$, $p\ge 7$ an odd prime such that $4\mathrel{\not|}(p-1)$, has exactly two orbital digraphs $\Gamma_1$, $\Gamma_2$, such that Aut$(\Gamma_i)$ admits a complete block system ${\cal B}$ of $p+1$ blocks of size $2$, $i = 1,2$, with the following properties: the action of Aut$(\Gamma_i)$ on the blocks of ${\cal B}$ is nonsolvable, doubly-transitive, but not a symmetric group, and the subgroup of Aut$(\Gamma_i)$ that fixes each block of ${\cal B}$ 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 $2^{k+1}$. In this case, these digraphs are non-normal Cayley digraphs of the generalized quaternion group of order $2^{k+1}$.