# List of small groups

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List of small groups

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.

Glossary

* Z"n": the cyclic group of order "n" (often the notation "C""n" is used, or Z/"n"Z).
* Dih"n": the dihedral group of order 2"n" (often the notation "D""n" or "D"2"n" is used )
* "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 "n"!/2 even permutations of "n" elements.
* Dic"n": the dicyclic group of order 4"n".

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" &times; "H" stands for the direct product of the two groups; "G""n" denotes the direct product of a group with itself "n" times. "G" &#x22CA; "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 non-trivial 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", where "n" is prime.) We use the equality sign ("=") to denote 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 multiple isomorphic subgroups, their number is indicated in parentheses.

List of small abelian groups

The finite abelian groups are easily classified: they are cyclic groups, or direct products thereof; see abelian groups.

List of small non-abelian groups

OrderGroupSubgroupsPropertiesCycle Graph
6"S"3 = Dih3Z3 , Z2 (3) the smallest non-abelian group
8
Dih4
Z4, Z22 (2) , Z2 (5)
Quaternion group, "Q"8 = Dic2 Z4 (3), Z2 the smallest Hamiltonian group
10Dih5 Z5 , Z2 (5)
12 Dih6 = Dih3 &times; Z2Z6 , Dih3 (2) , Z22 (3) , Z3 , Z2 (7)
"A"4 Z22 , Z3 (4) , Z2 (3) smallest group demonstrating that a group need not have a subgroup of every order that divides the group's order: no subgroup of order 6 (See Lagrange's theorem and the Sylow theorems.)
Dic3 = Z3 &#x22CA; Z4Z2, Z3, Z4 (3), Z6
14 Dih7 Z7, Z2 (7)
16 [Wild, Marcel. " [http://math.sun.ac.za/~wild/Marcel%20Wild%20-%20Home%20Page_files/Groups16AMM.pdf The Groups of Order Sixteen Made Easy] ", American Mathematical Monthly, Jan 2005] Dih8Z8, Dih4 (2), Z22 (4), Z4, Z2 (9)
Dih4 &times; Z2Dih4 (2), Z4 &times; Z2, Z23 (2), Z22 (7), Z4 (2), Z2 (11)
Generalized quaternion group, "Q"16 = Dic4
"Q"8 &times; Z2 Hamiltonian
The order 16 quasidihedral group
The order 16 modular group
Z4 &#x22CA; Z4
The group generated by the Pauli matrices
"G"4,4 = Z22 &#x22CA; Z4

mall groups library

The group theoretical computer algebra system GAP contains the "Small Groups library" which provides access to descriptions of the groups of "small" order. 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 (423 164 062 groups, the ones of order 1024 had to be skipped, there are alone 49 487 365 422 (up to isomorphism) 2-groups of order 1024.);
* those of order 55 and 74 (92 groups);
* those of order "q""n"&times;"p" where "q""n" divides 28, 36, 55 or 74 and "p" is an arbitrary prime which differs from "q";
* those whose order factorises into at most 3 primes.It contains explicit descriptions of the available groups in computer readable format.

The library has been constructed and prepared by Hans Ulrich Besche, Bettina Eick and Eamonn O'Brien; see http://www.tu-bs.de/~hubesche/small.html .

ee also

*Small Latin squares and quasigroups

* [http://www.math.usf.edu/~eclark/algctlg/small_groups.html Small groups]

References

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