Let $G_n=\mathbb{Z}_n\times \mathbb{Z}_n$ for $n\geq 4$ and $S=\{(i,0),(0,i),(i,i): 1\leq i \leq n-1\}\subset G_n$. Define $Γ(n)$ to be the Cayley graph of $G_n$ with respect to the connecting set $S$. It is known that $Γ(n)$ is a strongly regular graph with the parameters $(n^2,3n-3,n,6)$ \cite{19}. Hence $Γ(n)$ is a distance regular graph. It is known that every distance transitive graph is distance regular, but the converse is not true. In this paper, we study some algebraic properties of the graph $Γ(n)$. Then by determining the automorphism group of this family of graphs, we show that the graphs under study are not distance transitive.