The Ljubljana graph is a graph on 112 vertices that is the third smallest cubic semisymmetric graph. It was discovered by Brouwer et al. (1993) and rediscovered
by Conder et al. (2002), but appears to have been known by R. M. Foster
based on the comment, "R. M. Foster (private communication) has found
an edge- but not vertex-transitive cubic graph (with 112 vertices) whose girth (equal
to 10) is not a multiple of 4" appearing in Bouwer (1972).

It is illustrated above in a number of embeddings having order-2 LCF
notation.

The only cubic semisymmetric graphs on smaller numbers of vertices are the Gray graph on 54 vertices and the Iofinova-Ivanov
graph on 110 vertices.

Bouwer, I. A. "On Edge But Not Vertex Transitive Regular Graphs." J. Combin. Th. Ser. B12, 32-40, 1972.Brouwer,
A. E.; Dejter, I. J.; and Thomassen, C. "Highly Symmetric Subgraphs
of Hypercubes." J. Algebraic Combinat.2, 25-29, 1993.Conder,
M.; Malnič, A.; Marušič, D.; Pisanski, T.; and Potočnik, P.
"The Ljubljana Graph." 2002. http://citeseer.ist.psu.edu/conder02ljubljana.html.