Automorphism groups of a class of cubic Cayley graphs on symmetric groups

Let $S_n$ denote the symmetric group of degree $n$ with $n\ge 3$. Set $S=\{c_n=(12\dots n),c_n^{-1},(12)\}$. Let ${\mathrm{\Gamma }}_n=\mathrm{C}\mathrm{a}\mathrm{y}(S_n,S)$ be the Cayley graph on $S_n$ with respect to $S$. In this paper, we show that ${\mathrm{\Gamma }}_n$ ($n\ge 13$) is a normal Cayley graph, and that the full automorphism group of ${\mathrm{\Gamma }}_n$ is equal to $$\mathrm{A}\mathrm{u}\mathrm{t}$(\Gamma_n)=R(S_n)\rtimes \langle$\mathrm{I}\mathrm{n}\mathrm{n}$(\phi)\rangle\cong S_n\rtimes $\mathbb{Z}$_2$,where$R(S_n)$istherightregularrepresentationof$S_n$,$\phi=(1\ 2)(3\ n)(4\ n-1)(5\ n-2)\cdots$ $(\in S_n)$, and $\mathrm{I}\mathrm{n}\mathrm{n}(\varphi )$ is the inner isomorphism of $S_n$ induced by $\varphi$.