List of small groups

The following list in mathematics contains the finite groups of small order up to group isomorphism.

(sequence in the OEIS)

For labeled groups, see  .

Each group is named by their Small Groups library index as G_oⁱ, where o is the order of the group, and i is the index of the group within that order.

Common group names:

The notations Z_n and Dih_n have the advantage that point groups in three dimensions C_n and D_n do not have the same notation. There are more isometry groups than these two, of the same abstract group type.

The notation G × H 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 n < 60, the simple groups are precisely the cyclic groups Z_n, 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.

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