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 drawings. The map has 168 triangular faces.
Its automorphism group has group
order 1008 (Bokowski and Cuntz 2018).
The following cyclic construction organizes an edge list and two-half decomposition supplied by E. Pegg, Jr. (pers. comm., May 20-21,
2025). Take all subscripts modulo 7 and write 
for the undirected edge
with endpoints 
and 
.
The letters are simply orbit labels. One half with 36 vertices
consists of a fixed vertex 
and five families 
, 
, 
, 
, and 
with seven vertices each,
with one vertex in each family for every 
modulo 7. Simultaneously replacing every subscript 
by 
is an order-7 graph automorphism
of the half. For every 
, add the edges 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, and 
. Take a disjoint isomorphic primed copy on 
, 
, 
, 
, 
, and 
. Join the two halves, for every 
, by the edges 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, and 
. The resulting graph
is the Fricke-Macbeath graph.
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.