The latest Visual Insight post at the American Mathematical
Society website discusses group actions on the McGee graph,
pictured as 24 points arranged in a circle that are connected
by 36 symmetrically arranged edges.
Wikipedia remarks that …
"The automorphism group of the McGee graph
is of order 32 and doesn't act transitively upon
its vertices: there are two vertex orbits of lengths
8 and 16."
The partition into 8 and 16 points suggests, for those familiar
with the Miracle Octad Generator and the Mathieu group M24,
the following exercise:
Arrange the 24 points of the projective line
over GF(23) in a circle in the natural cyclic order
( ∞, 1, 2, 3, … , 22, 0 ). Can the McGee graph be
modeled by constructing edges in any natural way?
In other words, if the above set of edges has no
"natural" connection with the 24 points of the
projective line over GF(23), does some other
set of edges in an isomorphic McGee graph
have such a connection?
Update of 9:20 PM ET Sept. 20, 2015:
Backstory: A related question by John Baez
at Math Overflow on August 20.