Let G be a simple labeled graph with 12 vertices consisting of exactly two connected components which are path graphs on 6 vertices.
I think there are 4 automorphisms. Reverse one path, reverse the other path, or reverse both paths or do nothing (the identity mapping). If I try to count the labeleings I get binomial(12,6)*(6!/2)^2 and this is 12!/4. Everything checks. Right?
However, I am looking at data in Mathematica that says that this graph has 322560 automorphisms. This would imply 1485 labelings. Is this nonsense?