1. The problem statement, all variables and given/known data Consider the set of operations in the plane that includes rotations by an angle about the origin and reflections about an axis through the origin. Find a matrix representation in terms of 2x2 matrices of the group of transformations (rotations plus reflections) that leaves the snowflake invariant. Is the group Abelian or non-Abelian? 2. Relevant equations R = [cosθ sinθ] [-sinθ cosθ] Reflection so that x → -x: [-1 0][x] = [-x] [0 1][y] [y] Reflection so that y → -y: [1 0][x] = [x] [0 -1][y] [-y] Abelian: A.B = B.A 3. The attempt at a solution I know that I will be rotating the regular hexagon by increments of 60° so that each time I rotate, the hexagon is invariant. I know that the regular hexagon has 2n = 12 different symmetries, so I will have 12 2x2 matrices in total. I began by constructing my rotated matrices (0°, 60°, 120°, 180°, 240°, and 300°). I know that I will have 6 axes of symmetry to reflect by . . . 3 that stretch from vertex to vertex and 3 that stretch from the midpoint on one side of the hexagon to the opposite. I think I am finding that this group will be non-Abelian, but I am not sure. I have calculated the 6 matrices for the rotations but am unsure of what to do next. If I reflect each rotation so that the +x values become -x I get 6 matrices, but those are only for the symmetries along the axes from vertex to vertex. This cannot be correct since I am leaving out the symmetries that exist from midpoint to midpoint. How do I correct my approach so that it includes these symmetries? I have a midterm exam in the class this material appears in tomorrow morning and I know for certain that a problem very similar to this one will be on my exam. I went to my professor for help and he explained the general method for solving it, but I need some extra guidance! I am stuck. Please help!