What algebraic structures are related to the McGee graph? Recall that an $(n,g)$-graph is a simple graph where each node has $n$ neighbors and the shortest cycle has length $g$, while an $(n,g)$-cage is $(n,g)$-graph with the minimum number of nodes.  
The McGee graph is the unique $(3,7)$-cage.  It looks like this:

Now, the unique $(3,6)$-cage is the Heawood graph, which has strong connections to the Fano plane and its symmetry group, $PGL(3,\mathbb{F}_2)$.  The unique $(3,8)$-cage is the Tutte–Coxeter graph, which has strong connections to the Cremona–Richmond configuration and its symmetry group, $S_6$.  These stories are fascinating and strongly parallel.
What about the McGee graph?  All I know is what I read:


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*McGee graph, Wikipedia.  


It has 24 nodes and 36 edges.  Its symmetry group has 32 elements — which 32-element group is it?  Its symmetry group doesn't act transitively on the nodes: there are two orbits, containing 16 and 8 nodes.
Is this graph associated to an interesting finite geometry, or not?
 A: This is not an answer, only an illustration.
I tried to find a maximally symmetric embedding into 3-space.
It might be viewed as an embedding into a solid torus, with two disjoint 8-cycles on the surface and 8 vertices in four $\boldsymbol\rangle\!\!\!\boldsymbol-\!\!\!\boldsymbol\langle$-like shapes inside the torus.
Motion realizes an automorphism of order 8 (image of the embedding coincides with itself after every $\frac\pi2$-turn; after $2\pi$ one gets a nontrivial involution).
$\ \hskip8em \ $
Many thanks to Simon Rose for immediately noticing that my first attempt was obviously rubbish.
A: Thanks to Gordon Royle's help, Greg Egan and I were able to work out some of the algebraic structures related to the McGee graph.  I explained them here:


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*John Baez, The McGee graph, Visual Insight, 15 September 2015.

A: certainly Sage(math) can easily compute the structure of this group:
sage: g=graphs.McGeeGraph()
sage: a=g.automorphism_group()    
sage: a.structure_description()
'(C2 x D4) : C2'

(Sage makes an unusual choice to denote the dihedral group $D_{2k}$ of order $2k$ as $D_k$...)
A: 
Edited 27 September 2016
(An earlier version of this answer incorrectly claimed to show that the genus of the McGee graph was 3.  In fact, the genus of the McGee graph is 2.  Thanks to Gordon Royle, who pointed out this mistake.)
The image above shows how to embed the McGee graph into a surface of genus 2.  There are 10 oriented cycles here (8 of length 7, and 2 of length 8) which contain each edge twice with opposite orientations.  So the Euler characteristic of the oriented surface in which the graph is embedded here is $\chi=24-36+10=-2$, and its genus is $g=2$.
We can prove that this is the minimum genus as follows:
Suppose we've embedded the graph in an orientable surface, so that the surface is decomposed into $F$ regions.  These regions might not all be simply connected, so in general we will say that each has $b_i$ connected components to its boundary, and the $j$th connected component of the boundary of the $i$th region is comprised of $n_{i,j}$ edges of the graph, forming a cycle of the graph.
Since each of the 36 edges of the graph appears twice in these boundaries, we have:
$$\sum_{i=1}^F \sum_{j=1}^{b_i} n_{i,j} = 2 \times 36 = 72$$
But the smallest cycles in the graph have length 7, so:
$$7 \sum_{i=1}^F b_i \le 72$$
Since the $b_i$ are integers, it follows that:
$$\sum_{i=1}^F b_i \le 10$$
If the total number of holes in all the regions is $H$, we have:
$$F + H = \sum_{i=1}^F b_i \le 10$$
A division of the surface into $F$ regions with a total of $H$ holes can be converted into a division into $F$ simply connected regions by adding $H$ additional edges, without changing the number of vertices (e.g. an annulus can be converted into a disk by adding one edge).  Counting these $H$ supplementary edges in addition to the $E$ edges of the graph, we have the Euler characteristic for an orientable surface of genus $g$:
$$V - (E+H) + F = 24 - 36 + F - H = 2 - 2g$$
$$g = 7 - \frac{1}{2}(F-H)$$
But we have:
$$F-H = F+H-2H \le 10-2H$$
$$g \ge 2+H$$
So the genus of the surface must be at least 2, and $g=2$ is only possible when $H=0$ and the surface is divided into $F=10$ simply connected regions.
There is also a nice, symmetrical embedding of this graph into a surface of genus 3:

The image above shows 8 oriented 9-gons, arranged so that pairs of oppositely-oriented edges from different 9-gons form the edges of the McGee graph.  So the Euler characteristic of the oriented surface in which the graph is embedded here is $\chi=24-36+8=-4$, and its genus is $g=3$.
Any dihedral symmetry of the original planar embedding of the graph just permutes these eight 9-gons among themselves.  The other graph automorphisms give different 9-gons on the planar embedding, for example:

The Riemann surface for the genus 3 embedding looks like this:

where edges and faces with the same numbers should be identified.  The complete symmetry group of this surface is the 16-element dihedral group, with the orientation-preserving symmetries giving an 8-element cyclic subgroup.
