Lewis Stiller asks about software to look at Cayley graphs and
group action graphs.

What he needs is the GRAPE package for computing with graphs and
groups, which runs under the GAP system. In fact, GRAPE 2.1 is
about to be released as a GAP share package.

For now, here is a sample GRAPE computation of a CAYLEY graph
for the symmetric group S4, and some calls to GRAPE functions
showing that this graph has diameter 4, girth 6, and an automorphism
group of order 144.

gap> G:=SymmetricGroup(4);
Group( (1,4), (2,4), (3,4) )
gap> T:=G.generators;
[ (1,4), (2,4), (3,4) ]
gap> Cayley:=Graph(G,Elements(G),OnRight,function(x,y) return y/x in T; end);
rec(
  isGraph := true,
  order := 24,
  group := Group( ( 1, 2)( 3, 7)( 4, 9)( 5,11)( 6,13)( 8,16)(10,19)(12,21)
    (14,24)(15,23)(17,22)(18,20), ( 1, 3)( 2, 5)( 4,10)( 6,14)( 7,11)( 8,17)
    ( 9,18)(12,22)(13,23)(15,24)(16,21)(19,20), ( 1, 4)( 2, 6)( 3, 8)( 5,12)
    ( 7,15)( 9,13)(10,17)(11,20)(14,22)(16,23)(18,21)(19,24) ),
  schreierVector := [ -1, 1, 2, 3, 2, 3, 1, 3, 1, 2, 1, 3, 1, 2, 3, 1, 2, 2,
      1, 3, 1, 2, 2, 1 ],
  adjacencies := [ [ 2, 3, 4 ] ],
  representatives := [ 1 ],
  names := [ (), (1,4), (2,4), (3,4), (1,2,4), (1,3,4), (1,4,2), (2,3,4),
      (1,4,3), (2,4,3), (1,2), (1,2,3,4), (1,3), (1,3,2,4), (1,3,4,2),
      (1,4,2,3), (2,3), (1,2,4,3), (1,4,3,2), (1,2)(3,4), (1,2,3),
      (1,4)(2,3), (1,3)(2,4), (1,3,2) ] )
gap> Diameter(Cayley);
4
gap> Girth(Cayley);
6
gap> Size(AutGroupGraph(Cayley));
144
gap> quit;

L.H.Soicher@qmw.ac.uk