TITLE: Has anyone seen this graph?
QUESTION [8 upvotes]: I recently constructed the graph shown below in the process of investigating some problems regarding line graphs and homomorphisms, and then happened to see it on wikipedia. I was wondering if anyone knew of this graph coming up anywhere else.

If it helps, I was partly inspired by this paper which shows that any graph with maximum degree 3 and circular chromatic index 4 must contain $K_4$ with one edge subdivided as a subgraph. Note that the graph in the link above is three copies of $K_4$ with one edge subdivided plus another vertex which is adjacent to the vertices that are subdividing the edges.
Thanks

REPLY [3 votes]: Any connected trivalent graph realises a Schreier coset graph of a subgroup of
the modular group. This yields a transitive permutation group of degree 16 x 3
since we expand each node into an oriented triangle. In the original graph, 
switching the ends of edges & rotating the oriented triangles provides 
generators of order 2 and order 3. Recall PSL(2,Z) = free product C2 * C3.