There are 4858 connected cubic edge-transitive graphs on up to 10000 vertices, and for each of the 22 amalgam types the number of graphs of order at most n grows like n^{Theta(log n)}.
Density of quotient orders in groups and applications to locally-transitive graphs
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We prove that the set of orders of finite quotients of a finitely generated group has natural density 0, 1/2 or 1, and characterise when each of these cases occurs. We apply this to show that the sets of orders of various families of symmetric graphs have natural density 0.
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Edge-transitive cubic graphs: Cataloguing and Enumeration
There are 4858 connected cubic edge-transitive graphs on up to 10000 vertices, and for each of the 22 amalgam types the number of graphs of order at most n grows like n^{Theta(log n)}.