Let $S_n$ denote the symmetric group of degree $n$ with $n\geq 3$. Set $S=\{c_n=(1\ 2\ldots \ n),c_n^{-1},(1\ 2)\}$. Let $Γ_n=\mathrm{Cay}(S_n,S)$ be the Cayley graph on $S_n$ with respect to $S$. In this paper, we show that $Γ_n$ ($n\geq 13$) is a normal Cayley graph, and that the full automorphism group of $Γ_n$ is equal to $\mathrm{Aut}(Γ_n)=R(S_n)\rtimes \langle\mathrm{Inn}(φ)\rangle\cong S_n\rtimes \mathbb{Z}_2$, where $R(S_n)$ is the right regular representation of $S_n$, $φ=(1\ 2)(3\ n)(4\ n-1)(5\ n-2)\cdots$ $(\in S_n)$, and $\mathrm{Inn}(φ)$ is the inner isomorphism of $S_n$ induced by $φ$.