It’s a probable unit distance graph.
It has 356 4-colorings, and is not 3-colorable.
Its adjacency matrix is:
0 0 0 0 0 0 0 0 1 0 0 1 1
0 0 0 0 0 1 0 1 1 1 0 0 1
0 0 0 1 0 1 0 0 1 0 1 0 0
0 0 1 0 0 0 1 1 1 0 0 0 0
0 0 0 0 0 0 1 0 0 0 1 1 0
0 1 1 0 0 0 0 1 0 0 1 1 0
0 0 0 1 1 0 0 1 0 0 1 0 1
0 1 0 1 0 1 1 0 0 0 0 1 0
1 1 1 1 0 0 0 0 0 0 0 0 1
0 1 0 0 0 0 0 0 0 0 0 0 1
0 0 1 0 1 1 1 0 0 0 0 0 1
1 0 0 0 1 1 0 1 0 0 0 0 0
1 1 0 0 0 0 1 0 1 1 1 0 0
It also contains as a subgraph the Moser spindle; one of the vertices of the Moser spindle subgraph is the degree 4 (in the Moser spindle) vertex that is uppermost in the figure. Five of the seven vertices in the Moser spindle together with 5 edges amongst these 5 vertices make up a pentagon with bilateral symmetry; by staring enough at the figure, it can be found by me without too much trouble:
