ON A CLASS OF VERTEX-TRANSITIVE DISTANCE-REGULAR COVERS OF COMPLETE GRAPHS, II

Let Gamma be an abelian antipodal distance-regular graph of diameter 3 with the following property: (*) Gamma has a transitive group (G) over tilde of automorphisms which induces a primitive almost simple permutation group (G) over tilde (Sigma) on the set Sigma of its antipodal classes. If permutation rank rk((G) over tilde (Sigma)) of (G) over tilde (Sigma) equals 2, then Gamma is arc-transitive; moreover, all such graphs are now known. The purpose of this paper is to describe the graphs Gamma with the property (*) in the case when rk((G) over tilde Sigma) = 3. According to the classification of primitive almost simple permutation groups of rank 3 the socle of the group (G) over tilde (Sigma) under the given condition is either a sporadic simple group, or an alternating group, or a simple group of exceptional Lie type, or a classical simple group. Earlier, we described the graphs Gamma provided that rk((G) over tilde (Sigma)) = 3 and the socle of (G) over tilde (Sigma) is a sporadic simple group. Here we study the cases when (i) the socle of the group (G) over tilde (Sigma) is an alternating group or (ii) vertical bar Sigma vertical bar <= 2500 and socle of (G) over tilde (Sigma) is a simple group of exceptional Lie type. We show that the family of non-bipartite graphs Gamma with the property (*) and rk((G) over tilde (Sigma)) = 3 in the alternating case is finite and limited to a small number of potential examples with vertical bar Sigma vertical bar is an element of {10, 28,120}, each of which is a covering of one of five certain distance-transitive Taylor graphs. For each given group (G) over tilde (Sigma) of degree vertical bar Sigma vertical bar <= 2500 of exceptional type, we essentially restrict the set of admissible parameters of Gamma.