I have realized that my previous message could cause confusion:

I computed the automorphism group of my 72-vertex graph, yielding a
permutation group of size 144 and degree 72. In order to determine
whether or not the graph is a Cayley graph, I need to determine
whether or not the group contains a TRANSITIVE subgroup of order 72
(not just any subgroup of order 72).

When I tried "LatticeSubgroups" on this group, I got the error
message as outlined in my previous mail.

Then I checked (using IdGroup) that my permutation group was
isomorphic to SmallGroup(144,32) and just decided to try
LatticeSubgroups on that group. Notice that this would not
in fact be enough to determine if the graph is Cayley or not,
because the permutation representation is crucial, and
we have lost this information by using SmallGroup(144,32).

However this is a moot point because LatticeSubgroups fails both
on the permutation representation and also the representation
returned by SmallGroup.

--
Dr. Gordon F Royle, http://www.cs.uwa.edu.au/~gordon, gordon@cs.uwa.edu.au
--