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)}.
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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)}.