Automorphisms of a distance-regular graph with intersection array $\{75,64,18,1;1,6,64,75\}$

A distance-regular graph $\Gamma$ with intersection array $\{115,96,30,1;1,10,96,175\}$ is an $AT4$-graph. The antipodal quotient $\Gamma'$ has parameters $(392,115,18,40)$, and its first and second neighborhoods of vertices are strongly regular with parameters $(115,18,1,3)$ and $(276,75,10,24)$. Moreover, the second neighborhood of any vertex in $\Gamma_2(u)$ has intersection array $\{75,64,18,1;1,6,64,75\}$ and is a $4$-cover of a strongly regular graph with parameters $(276,75,10,24)$. Earlier, Makhnev, Paduchikh, and Samoilenko found possible automorphisms of a graph with parameters $(392,115,18,40)$ and of a graph with intersection array $\{115,96,30,1;1,10,96,175\}$. In this paper we find automorphisms of a graph with intersection array $\{75,64,18,1;1,6,64,75\}$. It is proved that the automorphism group of this graph acts intransitively on the set of its antipodal classes.