A list of 4-valent 2-arc-transitive graphs and finite faithful amalgams of index (4, 2)

An ss-arc in a simple graph ΓΓ is an (s+1)(s+1)-tuple of vertices of ΓΓ in which every two consecutive vertices are adjacent and every three consecutive vertices are pairwise distinct. A graph ΓΓ is said to be 2-arc-transitive if the automorphism group Aut(Γ) acts transitively on the set of 2-arcs of ΓΓ. It is shown that there are exactly 70 simple connected 2-arc-transitive 4-valent graphs on no more than 512 vertices. A description of these graphs as coset graphs is given, and some basic graph theoretical properties are computed. The list is obtained by first determining all finite faithful amalgams of index (4,2)(4,2), and then using a computer implementation of a small index subgroups algorithm.