Three local actions in 6-valent arc-transitive graphs

It is known that there are precisely three transitive permutation groups of degree 6 that admit an invariant partition with three parts of size 2 such that the kernel of the action on the parts has order 4; these groups are called A 4 ( 6 ), S 4 ( 6 d ) and S 4 ( 6 c ). For each L is an element of { A 4 ( 6 ) , S 4 ( 6 d ) , S 4 ( 6 c ) }, we construct an infinite family of finite connected 6-valent graphs { Gamma n } n is an element of N and arc-transitive groups G n <= Aut ( Gamma n ) such that the permutation group induced by the action of the vertex-stabiliser ( G n ) v on the neighbourhood of a vertex v is permutation isomorphic to L, and such that divide ( G n ) v divide is exponential in divide V ( Gamma n ) divide . These three groups were the only transitive permutation groups of degree at most 7 for which the existence of such a family was undecided. In the process, we construct an infinite family of cubic 2-arc-transitive graphs such that the dimension of the 1-eigenspace over the field of order 2 of the adjacency matrix of the graph grows linearly with the order of the graph.