Multiplication Table of C3V and P3 Symmetry Groups

[tableshark]

Can one set up a multiplication table for the symmetry group C3V of the equilateral
triangle.
Then show that it is identical to that of the permutation group P3.

I need some clarification...


What about a matrix representation (2x2) for these groups?
→ Here was thinking to use Cartesian coordinate
system in the xy plane, with origin at the center of the triangle.

 

[STEMucator]

I believe this will help answer your question about the symmetry group of the equilateral triangle:

http://dogschool.tripod.com/trianglegroup.html

The symmetry group of the equilateral triangle is indeed isomorphic to the dihedral group of order six (degree three) $D_3$.

It is also isomorphic to $SL(2, 2)$ I believe.