Cayley Graph of the Symmetry Group of the Triangular Prism 
Draw the Cayley Graph of the symmetry group of the triangular Prism. 

I am having a difficult time with this question. So far I know that the symmetry group has order 12, and also the symmetry group is D_3h. I just cannot seem to wrap my head around how to draw the cayley graph for this. Any help would be great. Thanks in advance.
 A: Why do you have four generators?  I would think three is natural:  rotation of 120 degrees, reflection about a plane parallel to the triangle, reflection about a plane orthogonal to the triangle.
Now write down the 12 elements of the group that you have in some organized way and draw edges between them when multiplication with a generator from the right gets you from one element to the other.  Label this edge with the corresponding generator.  That's your Caley graph.  (Each element should have three outgoing edges and three incoming edges.)
A: You say you work better when you have permutations as elements. Well, we can work with permutations. Draw the prism and name its vertices 1 through 6 in some way. Now figure out the reflections and rotations; each one maps the set of vertices onto itself, so you get a permutation, and that permutation uniquely defines the symmetry of the prism. From this point onwards you may forget about the geometry and just work with the permutations. 
Now, the generators: I'm not sure what you mean by "the first thing to do is get the generators chosen. I know there are 4..." - a group can be generated by more than one set of generators, and you can choose any set you like I guess if the problem doesn't specify one (you can point out to the prof, TA or author that "the Cayley graph of a group" is not a well defined concept). Now that you have permutations to work with, you may find it easier to pick some generating set and then draw the edges from each group element to its product by each generator.
