On a class of vertex-transitive distance-regular covers of complete graphs

In this paper, we investigate the problem of classification of abelian antipodal distance-regular graphs $\Gamma$ of diameter three with the following property $(*)$: there is a vertex-transitive group of automorphisms $G$ of $\Gamma$ which induces an almost simple primitive permutation group $G^{\Sigma}$ on the set $\Sigma$ of antipodal classes of $\Gamma$. This problem has been recently solved in the case when the permutation rank $\mathrm{rk}(G^{\Sigma})$ of $G^{\Sigma}$ equals $2$ (which implies classification of all arc-transitive representatives). Here we start to study the next case $\mathrm{rk}(G^{\Sigma})=3$. We elaborate a method of reduction to minimal quotients of $\Gamma$, which gives us a base for a classification scheme that depends on a type of such quotient. By analysing equitable partitions of $\Gamma$ which arise as collections of orbits of some subgroups of $G$, we obtain several strong restrictions on spectra and parameters of $\Gamma$ as well as a description of its minimal quotients. This allows us to settle the case when the socle of $G^{\Sigma}$ is a sporadic simple group.