List of small groups

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 G_''o''^''i'', 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/''n''Z).
* Dih_''n'': the dihedral group of order 2''n'' (often the notation D_''n'' or D_2''n'' is used )
** K₄: the Klein four-group of order 4, same as and 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 , and order ''n''!/2 otherwise.
* Dic_''n'' or Q_4''n'': the dicyclic group of order 4''n''.
** 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 denotes the direct product of the two groups; ''G''^''n'' 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 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.

Angle brackets 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 of nonisomorphic abelian groups of orders ''n'' = 1, 2, ... are
: 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 5, 1, 2, 1, 2, ...
For labeled abelian groups, see .

List of small non-abelian groups

The numbers of non-abelian groups, by order, are counted by . However, many orders have no non-abelian groups. The orders for which a non-abelian group exists 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, ...

Classifying groups of small order

Small groups of prime power order ''p''^''n'' are given as follows:
*Order ''p'': The only group is cyclic.
*Order ''p''²: There are just two groups, both abelian.
*Order ''p''³: 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 ''p''² by a cyclic group of order ''p''. The other is the quaternion group for and a group of exponent ''p'' for .
*Order ''p''⁴: 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
*Order 60: The alternating group A₅.

The smallest order for which it is ''not'' known how many nonisomorphic groups there are is 2048 = 2¹¹.

Small Groups Library

The GAP computer algebra system contains a package called 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:Hans Ulrich Besch
The Small Groups library

* those of order at most 2000 except for order 1024 ( groups in the library; the ones of order 1024 had to be skipped, as there are an additional nonisomorphic 2-groups of order 1024);
* those of cubefree order at most 50000 (395 703 groups);
* those of squarefree order;
* those of order ''p''^''n'' for ''n'' at most 6 and ''p'' prime;
* those of order ''p''⁷ for ''p'' = 3, 5, 7, 11 (907 489 groups);
* those of order ''pq''^''n'' where ''q''^''n'' 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 (not necessarily distinct).

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

The smallest order for which the SmallGroups library does not have information is 1024.

See also

*Classification of finite simple groups
*Composition series
*List of finite simple groups
* Number of groups of a given order
* Small Latin squares and quasigroups
*Sylow theorems

Notes

References

*, Table 1, Nonabelian groups order<32.
*

External links

* Particular groups in the Group Properties Wiki
* Groups of given order
* {{cite web
, first1 = H. U.
, last1 = Besche
, last2 = Eick
, first2 = B.
, last3 = O'Brien
, first3 = E.
, url = http://www.icm.tu-bs.de/ag_algebra/software/small/
, title = small group library
, url-status = dead
, archive-url = https://web.archive.org/web/20120305020857/http://www.icm.tu-bs.de/ag_algebra/software/small/
, archive-date = 2012-03-05

GroupNames database

Groups that are small
Groups that are small
Finite groups