Abstract
We study (G,2)-arc-transitive graphs for innately transitive permutation groups G such that G can be embedded into a wreath product SymΓwrSℓ acting in product action on Γℓ. We find two such connected graphs: the first is Sylvester's double six graph with 36 vertices, while the second is a graph with 1202 vertices whose automorphism group is AutSp(4,4). We prove that under certain conditions no more such graphs exist.
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