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The following list in mathematics contains the finite groups of small order up to group isomorphism.
The list can be used to determine which known group a given finite group G is isomorphic to: first determine the order of G, then look up the candidates for that order in the list below. If you know whether G is abelian or not, some candidates can be eliminated right away. To distinguish between the remaining candidates, look at the orders of your group's elements, and match it with the orders of the candidate group's elements.
Contents
Glossary
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 stands for the direct product of the two groups; G^{n} denotes the direct product of a group with itself n times. G 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 nontrivial 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 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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