TITLE: What algebraic structures are related to the McGee graph?
QUESTION [20 upvotes]: 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:

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?

REPLY [3 votes]: 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 $D_k$...)