Three local actions in 6‐valent arc‐transitive graphs

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