  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 $\Gamma(n)$ to
be the Cayley graph of $G_n$ with respect to the connecting set $S$. It is
known that $\Gamma(n)$ is a strongly regular graph with the parameters
$(n^2,3n-3,n,6)$ \cite{19}. Hence $\Gamma(n)$ 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 $\Gamma(n)$. Then by determining the automorphism group of this family of
graphs, we show that the graphs under study are not distance transitive.
