Prove that the following graph G is 5-arc-transitive, but not 6-arc-transitive. Let, S be a 6-set (1,2,3,4,5,6)

The vertex set of G, contains a vertex corresponding to each 2-subset of S eg. (1,2). It also contains a vertex corresponding to each partition of S into 2-subsets, eg. {(1,2),(3,4),(5,6)}. There are 15 vertices of each type, for a total of 30 vertices.

The edge set of G, is such that no vertex corresponding to a 2-subset is adjacent to another vertex corresponding to a 2-subset, and no vertex corresponding to a partition is adjacent to another vertex corresponding to a partition. However, each vertex corresponding to a 2-subset is adjacent to each vertex corresponding to a partition that contains it. There should be 45 such edges in total.