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

An s -arc in a simple graph Γ is an ( s + 1 ) -tuple of vertices of Γ in which every two consecutive vertices are adjacent and every three consecutive vertices are pairwise distinct. A graph Γ is said to be 2-arc-transitive if the automorphism group Aut ( Γ ) acts transitively on the set of 2-arcs of Γ . 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 ) , and then using a computer implementation of a small index subgroups algorithm.