The following list in mathematics contains the finite groups of small order up to group isomorphism. Counts For n = 1, 2, … the number of nonisomorphic groups of order n is 1, 1, 1, 2, 1, 2, 1, 5, 2, 2, 1, 5, 1, 2, 1, 14, 1, 5, 1, 5, ... For labeled groups, see . Glossary Each group is named by their Small Groups library as Goi, where o is the order of the group, and i is the index of the group within that order. Common group names: 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 ) K4: the Klein four-group of order 4, same as and Dih2. Sn: the symmetric group of degree n, containing the n! permutations of n elements. An: the alternating group of degree n, containing the even permutations of n elements, of order 1 for , and order n!/2 otherwise. Dicn or Q4n: the dicyclic group of order 4n. Q8: the quaternion group of order 8, also Dic2. 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 denotes the direct product of the two groups; Gn denotes the direct product of a group with itself n times. G ⋊ H denotes a semidirect product where H acts on G; this may also depend on the choice of action of H on G. Abelian and simple groups are noted. (For groups of order , 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 several isomorphic subgroups, the number of such subgroups is indicated in parentheses. Angle brackets <relations> show the presentation of a group. List of small abelian groups The finite abelian groups are either cyclic groups, or direct products thereof; see Abelian group. The numbers