Automorphisms of an AT4(4,4,2)-Graph and of the Corresponding Strongly Regular Graphs

Makhnev, Paduchikh, and Khamgokova gave a classification of distance-regular locally GQ(5, 3)-graphs. In particular, there arises an AT 4(4, 4, 2)-graph on 644 vertices with intersection array {96, 75, 16, 1; 1, 16, 75, 96}. The same authors proved that an AT 4(4, 4, 2)-graph is not a locally GQ(5, 3)-graph. However, the existence of an AT 4(4, 4, 2)-graph that is a locally pseudo-GQ(5, 3)-graph is unknown. The antipodal quotient of an AT 4(4, 4, 2)-graph is a strongly regular graph with parameters (322, 96, 20, 32). These two graphs are locally pseudo-GQ(5, 3)-graphs. We find their possible automorphisms. It turns out that the automorphism group of a distance-regular graph with intersection array {96, 75, 16, 1; 1, 16, 75, 96} acts intransitively on the set of its antipodal classes.