I have been told by an authority that he believed that the size of the
automorphism group of the Doyle graph is 54.  Unfortunately I could
not get definitive confirmation of this value, nor an explanation of
the answer from the authority.  Please provide a sketch proof or
explanation of why the value is 54, in terms of the generators and
relations that define this graph (or the correct figure if it's not
54).  I am not looking for a rigorous proof, really an explanation. 
More specifically, I know how to get the size of the automorphism
group from the size of the orbits.  What I don't know is how to get
the size of the orbits from the generators and relations (or perhaps
there is some better way to calculate the size of the automorphism
group).

I believe this is fairly elementary group theory, but my group theory
background is spotty.

Some information on the Doyle graph is here:
http://mathworld.wolfram.com/DoyleGraph.html

The definition of the Doyle graph in terms of group generators and
relations is here (see p. 2)
http://math.dartmouth.edu/~doyle/docs/bouwer/bouwer.pdf