A graph with intersection array {18, 15, 1; 1, 5, 18} is not vertex-symmetric

A.A. Makhnev and V.P. Burichenko found possible intersection arrays of distance-regular locally cyclic graphs with at most 1000 vertices. They proposed a program for studying arc-transitive graphs with these intersection arrays. The neighborhood of a vertex in such a graph is the union of isolated polygons. We study automorphisms of a hypothetical distance-regular graph with intersection array {18, 15, 1; 1, 5, 18}. In particular, we prove that the automorphism group of this graph acts intransitively on the vertex set.