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ABSTRACT. The permutation representations in the title are all determined, and no surprises are found to occur.

A distance-transitive antipodal cover of a complete graph K n possesses an automorphism group that acts 2-transitively on the fibres. The classification of finite simple groups implies a classification of finite 2-transitive permutation groups, and this allows us to determine all possibilities for such a graph. Several new infinite families of distance-transitive graphs are constructed.

The Johnson graph J(v, k) has, as vertices, the k-subsets of a v-set V and as edges the pairs of k-subsets with intersection of size k − 1. We introduce the notion of a neighbour-transitive code in J(v, k). This is a vertex subset Γ such that the subgroup G of graph automorphisms leaving Γ invariant is transitive on both the set Γ of 'codewords' and also the set of 'neighbours' of Γ, which are the non-codewords joined by an edge to some codeword. We classify all examples where the group G is a subgroup of the symmetric group Sym (V) and is intransitive or imprimitive on the underlying v-set V. In the remaining case where G ≤ Sym (V) and G is primitive on V, we prove that, provided distinct codewords are at distance at least 3, then G is 2-transitive on V. We examine many of the infinite families of finite 2-transitive permutation groups and construct surprisingly rich families of examples of neighbour-transitive codes. A major unresolved case remains.

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