The Fricke-Macbeath graph, also called the Macbeath graph, is the 1-skeleton of the Hurwitz map 
on the Fricke-Macbeath
curve, which has genus 7. It is a septic
graph with 72 vertices and 252 edges,
illustrated above in a number of embeddings. The
map has 168 triangular faces. Its automorphism
group has group order 1008 (Bokowski and Cuntz
2018).
The Fricke-Macbeath graph is implemented in the Wolfram Language as GraphData["FrickeMacbeathGraph"].
The local graph at every vertex is the cycle graph 
.
The graph is vertex-transitive, edge-transitive,
and arc-transitive, as well as Hamiltonian
and Hamilton-connected.
It has graph genus 7, girth 3, graph radius 5, graph diameter 5, chromatic number 5, edge chromatic number 7, clique number 3, and independence number 18. Its characteristic polynomial is
 |
of Genus 7: A Polyhedral Realization."
Art Discrete Appl. Math. 1, #P1.02, 2018.