Zn: the cyclic group of order n (the notation Cn is also used; it is isomorphic to the additive group of Z/nZ). Dihn: the dihedral group of order 2n (often the notation Dn or D2n is used ) Sn: the symmetric group of degree n, containing the n! permutations of n elements. An: the alternating group of degree n, containing the n!/2 even permutations of n elements. Dicn: the dicyclic group of order 4n. The notations Zn and Dihn have the advantage that point groups in three dimensions Cn and Dn do not have the same notation. There are more isometry groups than these two, of the same abstract group type. The notation G × H stands for the direct product of the two groups; Gn denotes the direct product of a group with itself n times. G \rtimes H stands for a semidirect product where H acts on G; where the particular action of H on G is omitted, it is because all possible non-trivial actions result in the same product group, up to isomorphism. Abelian and simple groups are noted. (For groups of order n < 60, the simple groups are precisely the cyclic groups Zn, for prime n.) The equality sign ("=") denotes isomorphism. The identity element in the cycle graphs is represented by the black circle. The lowest order for which the cycle graph does not uniquely represent a group is order 16. In the lists of subgroups, the trivial group and the group itself are not listed. Where there are multiple isomorphic subgroups, their number is indicated in parentheses. |

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