= e; so each of the above actions has an inverse. By inspection, we can also determine associativity and closure (two of the necessary group axioms); note for example that (ab)a = a(ba) = aba, and (ba)b = b(ab) = bab. The group is non-abelian since, for example, . Since it is built up from the basic actions a and b, we say that the set generates it. The group has presentation , also written or , also written where a and b are swaps and is a cyclic permutation. Note that the second presentation means that the group is a Coxeter group. (In fact, all dihedral and symmetry groups are Coxeter groups.) Summary of group operations With the generators a and b, we define the additional shorthands , and , so that a, b, c, d, e, and f are all the elements of this group. We can then summarize the group operations in the form of a Cayley table: Note that non-equal non-identity elements only commute if they are each other's inverse. Therefore, the group is centerless, i.e., the center of the group consists only of the identity element. Conjugacy classes We can easily distinguish three kinds of permutations of the three blocks, the conjugacy classes of the group: no change (), a group element of order 1 interchanging two blocks: (RG), (RB), (GB), three group elements of order 2 a cyclic permutation of all three blocks: (RGB), (RBG), two group elements of order 3 For example, (RG) and (RB) are both of the form (x y); a permutation of the letters R, G, and B (namely (GB)) changes the notation (RG) into (RB). Therefore, if we apply (GB), then (RB), and then the inverse of (GB), which is also (GB), the resulting permutation is (RG). Note that conjugate group elements always have the same order, but in general two group elements that have the same order need not be conjugate. Subgroups From Lagrange's theorem we know that any non-trivial subgroup of a group with 6 elements must have order 2 or 3. In fact the two cyclic permutations of all three blocks, with the identity, form a subgroup of order 3, index 2, and the swaps of two blocks, each with the identity, form three subgroups of order 2, index 3. The existence of subgroups of order 2 and 3 is also a consequence of Cauchy's theorem.