The first-mentioned is the alternating group A3. The left cosets and the right cosets of A3 coincide (as they do for any subgroup of index 2) and consist of A3 and the set of three swaps }. The left cosets of are: { (), (RG) } { (RB), (RGB) } { (GB), (RBG) } The right cosets of are: { (RG), () } { (RBG), (RB) } { (RGB), (GB) } Thus A3 is normal, and the other three non-trivial subgroups are not. The quotient group is isomorphic with C2. , a semidirect product, where H is a subgroup of two elements: () and one of the three swaps. This decomposition is also a consequence (particular case) of the Schur–Zassenhaus theorem. In terms of permutations the two group elements of are the set of even permutations and the set of odd permutations. If the original group is that generated by a 120°-rotation of a plane about a point, and reflection with respect to a line through that point, then the quotient group has the two elements which can be described as the subsets "just rotate (or do nothing)" and "take a mirror image". Note that for the symmetry group of a square, an uneven permutation of vertices does not correspond to taking a mirror image, but to operations not allowed for rectangles, i.e. 90° rotation and applying a diagonal axis of reflection. Semidirect products is if both φ(0) and φ(1) are the identity. The semidirect product is isomorphic to the dihedral group of order 6 if φ(0) is the identity and φ(1) is the non-trivial automorphism of C3, which inverses the elements. Thus we get: (n1, 0) * (n2, h2) = (n1 + n2, h2) (n1, 1) * (n2, h2) = (n1 − n2, 1 + h2) for all n1, n2 in C3 and h2 in C2. More concisely, for all n1, n2 in C3 and h1, h2 in C2. In a Cayley table: Note that for the second digit we essentially have a 2×2 table, with 3×3 equal values for each of these 4 cells. For the first digit the left half of the table is the same as the right half, but the top half is different from the bottom half. For the direct product the table is the same except that the first digits of the bottom half of the table are the same as in the