List of small groups

Counts

(sequence A000001 in the OEIS)

For labeled groups, see  A034383.

Glossary

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:

Z_n: the cyclic group of order n (the notation C_n is also used; it is isomorphic to the additive group of Z/nZ).

Dih_n: the dihedral group of order 2n (often the notation D_n or D_2n is used )

K₄: the Klein four-group of order 4, same as Z₂ × Z₂ or Dih₂.

S_n: the symmetric group of degree n, containing the n! permutations of n elements.

A_n: the alternating group of degree n, containing the even permutations of n elements, of order 1 for n = 0, 1, and order n!/2 otherwise.

Dic_n or Q_4n: the dicyclic group of order 4n.

Q₈: the quaternion group of order 8, also Dic₂.

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.

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.

List of small abelian groups

The finite abelian groups are either cyclic groups, or direct products thereof; see abelian groups.

(sequence A000688 in the OEIS)

For labeled Abelian groups, see  A034382.

List of small non-abelian groups

(sequence A060689 in the OEIS)

Order of non-abelian groups are

6, 8, 10, 12, 14, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30, 32, 34, 36, 38, 39, 40, 42, 44, 46, 48, 50, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 66, 68, 70, 72, 74, 75, 76, 78, 80, 81, 82, 84, 86, 88, 90, 92, 93, 94, 96, 98, 100, 102, 104, 105, 106, 108, 110, 111, 112, 114, 116, 117, 118, 120, 122, 124, 125, 126, 128, 129, 130, 132, 134, 135, 136, 138, 140, 142, ... (sequence A060652 in the OEIS)

Classifying groups of small order

Small groups of prime power order pⁿ are given as follows:

Order p: The only group is cyclic.

Order p²: There are just two groups, both abelian.

Order p³: There are three abelian groups, and two non-abelian groups. One of the non-abelian groups is the semidirect product of a normal cyclic subgroup of order p² by a cyclic group of order p. The other is the quaternion group for p = 2 and a group of exponent p for p > 2.

Order p⁴: The classification is complicated, and gets much harder as the exponent of p increases.

Most groups of small order have a Sylow p subgroup P with a normal p-complement N for some prime p dividing the order, so can be classified in terms of the possible primes p, p-groups P, groups N, and actions of P on N. In some sense this reduces the classification of these groups to the classification of p-groups. Some of the small groups that do not have a normal p complement include:

Order 24: The symmetric group S₄

Order 48: The binary octahedral group and the product S₄ × Z₂

Order 60: The alternating group A₅.

Small groups library

The group theoretical computer algebra system GAP contains the "Small Groups library" which provides access to descriptions of small order groups. The groups are listed up to isomorphism. At present, the library contains the following groups:

those of order at most 2000, except for order 1024 (423164062 groups in the library; the ones of order 1024 had to be skipped, as there are an additional 49487365422 nonisomorphic 2-groups of order 1024.);

those of cubefree order at most 50000 (395 703 groups);

those of squarefree order;

those of order pⁿ for n at most 6 and p prime;

those of order p⁷ for p = 3, 5, 7, 11 (907 489 groups);

those of order pqⁿ where qⁿ divides 2⁸, 3⁶, 5⁵ or 7⁴ and p is an arbitrary prime which differs from q;

those whose orders factorise into at most 3 primes (count with multiplicity).

It contains explicit descriptions of the available groups in computer readable format.

The smallest 10 orders for which the SmallGroups library does not have information are 1024, 2016, 2024, 2025, 2040, 2048, 2052, 2058, 2064, 2072.