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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.

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    By "triangular prism" you must mean "regular tetrahedron". To get a Cayley graph you also need to choose generators - does the homework question specify what those should be? Finally, what do you know about the elements of the group? Are you able to list them in some way?2011-09-28
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    No it is not the regular tetrahedron. It is a pentahedron, so the two trinagular faces are parallel. Yes, the first thing to do is to get the generators chosen. I know there are 4, and no I do not know all of the elements. That is partially my problem. I can draw the triangular prism and figure out the reflections and rotations. I just seem to work better with having permutations as elements when I draw cayley graphs. I am unable to do this at this point with this problem.2011-09-28
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    What is "D_3h"?2011-09-28
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    Is the "triangular prism" like a triangular peg? Two (equilateral?) triangles, one on top and one on the bottom, and three rectangular sides?2011-09-28
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    @Arturo: Yes, a prism over S is { (x,y,z) : (x,y) in S, 0 ≤ z ≤ 1 } or so. D_3h is literally the symmetry group of this prism. I think the D_3 means the dihedral group of order 6, acting planewise in its natural action as the symmetry group of the triangular cross-sections. The sub h means an additional generator is added, probably a horizontal reflection, meaning, it takes one triangular base to the opposite triangular base.2011-09-28
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    @Jack: Thanks! So the elements are the identity, the two rotations along a vertical axis, the three reflections along planes perpendicular to the $xy$ plane; and each of those followed by (or preceded by) a reflection about a plane parallel to the two triangular bases.2011-09-28
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    Yup, exactly. I think Stefan G's answer should give that the cayley graph is in fact a related geometric figure, but I haven't had time to make sure.2011-09-28
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    Thank you everyone for the help. I am sorry if I was unclear about anything.2011-09-28
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    @Arturo: it's [Schönflies notation](http://en.wikipedia.org/wiki/Schoenflies_notation) for crystallographic point groups. $D_{3h}$ would indeed be notation for the symmetry group of the triangular prism, as well as an equilateral triangle...2011-10-02

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