Structure of the automorphism group of the augmented cube graph

Abstract

\noindent The augmented cube graph $AQ_n$ is the Cayley graph of $\mathbb{Z}_2^n$ with respect to the set of $2n-1$ generators $\{e_1,e_2, \ldots,e_n, 00\ldots0011, 00\ldots0111, 11\ldots1111 \}$. It is known that the order of the automorphism group of the graph $AQ_n$ is $2^{n+3}$, for all $n \ge 4$. In the present paper, we obtain the structure of the automorphism group of $AQ_n$ to be \[ \Aut(AQ_n) \cong \mathbb{Z}_2^n \rtimes D_8~~(n \ge 4),\] where $D_8$ is the dihedral group of order 8. It is shown that the Cayley graph $AQ_3$ is non-normal and that $AQ_n$ is normal for all $n \ge 4$. We also analyze the clique structure of $AQ_4$ and show that the automorphism group of $AQ_4$ is isomorphic to that of $AQ_3$: \[ \Aut(AQ_4) \cong \Aut(AQ_3) \cong (D_8 \times D_8) \rtimes C_2.\] All the nontrivial blocks of $AQ_4$ are also determined.