Automorphisms of a distance regular graph with intersection array {48,35,9;1,7,40}

If a distance-regular graph Γ of diameter 3 contains a maximal locally regular 1-code perfect with respect to the last neighborhood, then Γ has an intersection array {a(p+1),cp,a+1;1,c,ap} or {a(p+1),(a+1)p,c;1,c,ap}, where a=a3, c=c2, p=p333 (Jurisic and Vidali). In the first case, Γ has an eigenvalue θ2=-1 and Γ3 is a pseudo-geometric graph for GQ(p+1,a). If c=a-1=q, p=q-2, then Γ has an intersection array {q2-1,q(q-2),q+2;1,q,(q+1)(q-2)}, q>6. The orders and subgraphs of fixed points of automorphisms of a hypothetical distance-regular graph with intersection array {48,35,9;1,7,40} (q=7) are studied in the paper. Let G=Aut(Γ) be an insoluble group acting transitively on the set of vertices of the graph Γ, K=O7(G), T¯ be the socle of the group G¯=G/K. Then T¯ contains the only component L¯, L¯ that acts exactly on K, L¯≅L2(7),A5,A6,PSp4(3) and for the full the inverse image of L of the group L¯ we have La=Ka×O7′(La) and |K|=73 in the case of L¯≅L2(7), |K|=74 otherwise.