This is the McGee graph. You’ll notice each vertex has 3 neighbors. Also, if you go around a cycle of edges, there will always be at least 7 edges in your path. The McGee graph’s claim to fame is that it has the fewest vertices possible for a graph with these two properties.
Besides the obvious symmetries – rotating it 1/8 of a turn, or flipping it over – there are some sneaky symmetries, like the one shown in this animation by Greg Egan [Gif 3] .
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