The following list in mathematics contains the finite groups of small order up to group isomorphism. Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ... In mathematics, a finite group is a group which has finitely many elements. ... In group theory, the term order is used in two closely related senses: the order of a group is its cardinality, i. ... Look up Up to on Wiktionary, the free dictionary In mathematics, the phrase up to xxxx indicates that members of an equivalence class are to be regarded as a single entity for some purpose. ... In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations. ...

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 2n (often the notation D_n is used, and sometimes D_2n )
• 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 4n.

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. In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element a (called a generator of the group) such that, when written multiplicatively, every element of the group is a power of a. ... This article may be confusing for some readers, and should be edited to enhance clarity. ... In mathematics, the symmetric group on a set X, denoted by SX or Sym(X), is the group whose underlying set is the set of all bijective functions from X to X, in which the group operation is that of composition of functions, i. ... In mathematics, the factorial of a natural number n is the product of all positive integers less than and equal to n. ... In mathematics, especially in abstract algebra and related areas, a permutation is a bijection from a finite set X onto itself. ... In mathematics an alternating group is the group of even permutations of a finite set. ... In mathematics, the permutations of a finite set (i. ... In group theory, a dicyclic group is a member of a class of groups which are formed by an extension of a group (generally a cyclic group) by a cyclic group of order 2 (the latter giving the name di-cyclic). ... A discrete point group in 3D is a finite symmetry group in 3D that leaves the origin fixed. ... In geometry and mathematical analysis, an isometry is a bijective distance-preserving mapping. ...

The notation G × H stands for the direct product of the two groups. 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. In mathematics, one can often define a direct product of objects already known, giving a new one. ... In mathematics, an abelian group, also called a commutative group, is a group (G, *) such that a * b = b * a for all a and b in G. Abelian groups are named after Niels Henrik Abel. ... In mathematics, a simple group is a group G such that G is not the trivial group and the only normal subgroups of G are the trivial group and G itself. ...

The identity element in the cycle graphs are represented by the black circle. The lowest order for which the cycle graph does not uniquely represent a group is order 16. In group theory a group cycle graph illustrates the various cycles of a group and is particularly useful in visualizing the structure of small finite groups. ...

In the lists of subgroups the trivial group and the group itself are not listed.

List of small non-abelian groups

See also the list of small abelian groups and the combined list below. In mathematics, an abelian group, also called a commutative group, is a group (G, *) such that a * b = b * a for all a and b in G. Abelian groups are named after Niels Henrik Abel. ...

Note that e.g. "3 × Z₂" means that there are 3 subgroups of type Z₂ (NOT a left coset of Z₂), while elsewhere the cross means direct product. In mathematics, one can often define a direct product of objects already known, giving a new one. ...

Order	Group	Subgroups	Properties	Cycle graph
6	S3 = Dih3	Z3 , 3 × Z2	the smallest non-abelian group
8	Dih4	Z4, 2 × Dih2 , 5 × Z2	non-abelian
	Quaternion group, Q8 = Dic2	3 × Z4 , Z2	non-abelian; the smallest non-abelian Hamiltonian group
10	Dih5	Z5 , 5 × Z2	non-abelian
12	Dih6 = Dih3 × Z2	Z6 , 2 × Dih3 , 3 × Dih2 , Z3 , 7 × Z2	non-abelian		edit
	A4	Z22, 4 × Z3, 3 × Z2	non-abelian; smallest group demonstrating that the converse of Lagrange's theorem is not true: no subgroup of order 6
	Dic3 = the semidirect product of Z3 and Z4, where Z4 acts on Z3 by inversion		non-abelian
14	Dih7	Z7 , 7 × Z2	non-abelian
16	Dih8	Z8 , 2 × Dih4 , 4 × Dih2 , Z4 , 9 × Z2	non-abelian		edit
	Dih4 × Z2	2 × Dih4 , Z4 × Z2 , 2 × Z23, 7 × Z22 , 2 × Z4 , 11 × Z2	non-abelian
	Generalized quaternion group, Q16 = Dic4		non-abelian
	Q8 × Z2		non-abelian
	The order 16 quasidihedral group		non-abelian
	The order 16 modular group		non-abelian
	The semidirect product of Z4 and Z4 where one factor acts on the other by inversion		non-abelian
	The group generated by the Pauli matrices		non-abelian
	G4,4		non-abelian

The smallest non-Abelian group has 6 elements. ... Cycle diagram for group D6 File links The following pages link to this file: List of small groups Cycle graph (group) Categories: User-created public domain images ... Some elementary examples of groups in mathematics are given on Group (mathematics). ... Cycle diagram for group D8 File links The following pages link to this file: List of small groups Cycle graph (group) Categories: User-created public domain images ... Cycle diagram of Q. Each color specifies a series of powers of any element connected to the identity element (1). ... In group theory, a non-abelian group G is called Hamiltonian if every subgroup of G is normal. ... Image File history File links GroupDiagramMiniQ8. ... Cycle diagram for group D10 File links The following pages link to this file: List of small groups Cycle graph (group) Categories: User-created public domain images ... Cycle diagram for group D12 File links The following pages link to this file: List of small groups Cycle graph (group) Categories: User-created public domain images ... In mathematics, most commonly, Lagranges theorem states that if G is a finite group and H is a subgroup of G, then the order (that is, the number of elements) of H divides the order of G. This can be shown using the concept of left cosets of H... Image File history File links GroupDiagramMiniA4. ... In abstract algebra, a semidirect product describes a particular way in which a group can be put together from two subgroups. ... Image File history File links GroupDiagramMiniX12. ... Cycle diagram for group D14 File links The following pages link to this file: List of small groups Cycle graph (group) Categories: User-created public domain images ... Image File history File links GroupDiagramMiniD16. ... Image File history File links GroupDiagramMiniC2D8. ... Cycle diagram of Q. Each color specifies a series of powers of any element connected to the identity element (1). ... Image File history File links GroupDiagramMiniQ16. ... Image File history File links GroupDiagramMiniC2Q8. ... In mathematics, the quasidihedral groups (also known as semidihedral groups) are groups with similar properties to the dihedral groups. ... Image File history File links GroupDiagramMiniQH16. ... In mathematics, the modular group Γ (Gamma) is a group that is a fundamental object of study in number theory, geometry, algebra, and many other areas of advanced mathematics. ... Image File history File links GroupDiagramMiniMOD16. ... In abstract algebra, a semidirect product describes a particular way in which a group can be put together from two subgroups. ... Image File history File links GroupDiagramMinix3. ... The Pauli matrices are a set of 2 × 2 complex Hermitian matrices developed by Pauli. ... Image File history File links GroupDiagramMiniPauli. ... Image File history File links GroupDiagramMiniG44. ...

Combined list

Order	Group	Subgroups	Properties	Cycle graph
1	trivial group = Z1 = S1 = A2	-	abelian; this and various other properties hold trivially
2	Z2 = S2 = Dih1	-	abelian, simple, the smallest non-trivial group
3	Z3 = A3	-	abelian, simple
4	Z4	Z2	abelian
	Klein four-group = Z2 × Z2 = Dih2	3 × Z2	abelian, the smallest non-cyclic group
5	Z5	-	abelian, simple
6	Z6 = Z2 × Z3	Z2 , Z3	abelian
	S3 = Dih3	Z3 , 3 × Z2	the smallest non-abelian group
7	Z7	-	abelian, simple
8	Z8	Z4 , Z2	abelian
	Z2 ×Z4	2 × Z4 , 3 ×Z2	abelian
	Z2 × Z2 × Z2 = Dih2 × Z2	7 × Z2 × Z2 , 7 × Z2	abelian
	Dih4	Z4, 2 × Dih2 , 5 × Z2	non-abelian
	Quaternion group, Q8 = Dic2	3 × Z4 , Z2	non-abelian; the smallest non-abelian Hamiltonian group
9	Z9	Z3	abelian
	Z3 × Z3	4 × Z3	abelian
10	Z10 = Z2 × Z5	Z5 , Z2	abelian
	Dih5	Z5 , 5 × Z2	non-abelian
11	Z11	-	abelian, simple
12	Z12 = Z4 × Z3	Z6 , Z4 , Z3 , Z2	abelian
	Z2 × Z6 = Z2 × Z2 × Z3 = Dih2 × Z3	2 × Z6, Z3 , 3 × Z2	abelian
	Dih6 = Dih3 × Z2	Z6 , 2 × Dih3 , 3 × Dih2 , Z3 , 7 × Z2	non-abelian		edit
	A4	Z22, 4 × Z3, 3 × Z2	non-abelian; smallest group demonstrating that the converse of Lagrange's theorem is not true: no subgroup of order 6
	Dic3 = the semidirect product of Z3 and Z4, where Z4 acts on Z3 by inversion		non-abelian
13	Z13	-	abelian, simple
14	Z14 = Z2 × Z7	Z7 , Z2	abelian
	Dih7	Z7 , 7 × Z2	non-abelian
15	Z15 = Z3 × Z5	Z5 , Z3	abelian
16	Z16	Z8 , Z4 , Z2	abelian
	Z24		abelian
	Z4 × Z22		abelian
	Z8 × Z2		abelian
	Z42		abelian
	Dih8	Z8 , 2 × Dih4 , 4 × Dih2 , Z4 , 9 × Z2	non-abelian		edit
	Dih4 × Z2	2 × Dih4 , Z4 × Z2 , 2 × Z23, 7 × Z22 , 2 × Z4 , 11 × Z2	non-abelian
	Generalized quaternion group, Q16 = Dic4		non-abelian
	Q8 × Z2		non-abelian
	The order 16 quasidihedral group		non-abelian
	The order 16 modular group		non-abelian
	The semidirect product of Z4 and Z4 where one factor acts on the other by inversion		non-abelian
	The group generated by the Pauli matrices		non-abelian
	G4,4		non-abelian

In mathematics, the term trivial is frequently used for objects (for examples, groups or topological spaces) that have a very simple structure. ... Cycle diagam for the group C1 File links The following pages link to this file: List of small groups Cycle graph (group) Categories: User-created public domain images ... Image File history File links GroupDiagramMiniC2. ... Cycle diagram for group C3 File links The following pages link to this file: List of small groups Cycle graph (group) Categories: User-created public domain images ... Cycle diagram for group C4 File links The following pages link to this file: List of small groups Cycle graph (group) Categories: User-created public domain images ... This article is about the mathematical group. ... Cycle diagram for group D4 File links The following pages link to this file: List of small groups Cycle graph (group) Categories: User-created public domain images ... Cycle diagram for group C5 File links The following pages link to this file: List of small groups Cycle graph (group) Categories: User-created public domain images ... Cycle diagram for group C6 File links The following pages link to this file: List of small groups Cycle graph (group) Categories: User-created public domain images ... The smallest non-Abelian group has 6 elements. ... Cycle diagram for group D6 File links The following pages link to this file: List of small groups Cycle graph (group) Categories: User-created public domain images ... Cycle diagram for group C7 File links The following pages link to this file: List of small groups Cycle graph (group) Categories: User-created public domain images ... In group theory, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group... Cycle diagram for group C8 File links The following pages link to this file: List of small groups Cycle graph (group) Categories: User-created public domain images ... Image File history File links GroupDiagramMiniC2C4. ... Cycle diagram for group C2xC2xC2 File links The following pages link to this file: List of small groups Cycle graph (group) Categories: User-created public domain images ... Some elementary examples of groups in mathematics are given on Group (mathematics). ... Cycle diagram for group D8 File links The following pages link to this file: List of small groups Cycle graph (group) Categories: User-created public domain images ... Cycle diagram of Q. Each color specifies a series of powers of any element connected to the identity element (1). ... In group theory, a non-abelian group G is called Hamiltonian if every subgroup of G is normal. ... Image File history File links GroupDiagramMiniQ8. ... Image File history File links GroupDiagramMiniC9. ... Cycle diagram for group C3xC3 File links The following pages link to this file: List of small groups Cycle graph (group) Categories: User-created public domain images ... Image File history File links GroupDiagramMiniC10. ... Cycle diagram for group D10 File links The following pages link to this file: List of small groups Cycle graph (group) Categories: User-created public domain images ... Image File history File links GroupDiagramMiniC11. ... Image File history File links GroupDiagramMiniC12. ... Image File history File links GroupDiagramMiniC2C6. ... Cycle diagram for group D12 File links The following pages link to this file: List of small groups Cycle graph (group) Categories: User-created public domain images ... In mathematics, most commonly, Lagranges theorem states that if G is a finite group and H is a subgroup of G, then the order (that is, the number of elements) of H divides the order of G. This can be shown using the concept of left cosets of H... Image File history File links GroupDiagramMiniA4. ... In abstract algebra, a semidirect product describes a particular way in which a group can be put together from two subgroups. ... Image File history File links GroupDiagramMiniX12. ... Image File history File links GroupDiagramMiniC13. ... Image File history File links GroupDiagramMiniC14. ... Cycle diagram for group D14 File links The following pages link to this file: List of small groups Cycle graph (group) Categories: User-created public domain images ... Image File history File links GroupDiagramMiniC15. ... Image File history File links GroupDiagramMiniC16. ... Cycle diagram for group C2xC2xC2xC2 File links The following pages link to this file: List of small groups Cycle graph (group) Categories: User-created public domain images ... Image File history File links GroupDiagramMiniC2x2C4. ... Image File history File links GroupDiagramMiniC2C8. ... Cycle diagram for group C4xC4 File links The following pages link to this file: List of small groups Cycle graph (group) Categories: User-created public domain images ... Image File history File links GroupDiagramMiniD16. ... Image File history File links GroupDiagramMiniC2D8. ... Cycle diagram of Q. Each color specifies a series of powers of any element connected to the identity element (1). ... Image File history File links GroupDiagramMiniQ16. ... Image File history File links GroupDiagramMiniC2Q8. ... In mathematics, the quasidihedral groups (also known as semidihedral groups) are groups with similar properties to the dihedral groups. ... Image File history File links GroupDiagramMiniQH16. ... In mathematics, the modular group Γ (Gamma) is a group that is a fundamental object of study in number theory, geometry, algebra, and many other areas of advanced mathematics. ... Image File history File links GroupDiagramMiniMOD16. ... In abstract algebra, a semidirect product describes a particular way in which a group can be put together from two subgroups. ... Image File history File links GroupDiagramMinix3. ... The Pauli matrices are a set of 2 × 2 complex Hermitian matrices developed by Pauli. ... Image File history File links GroupDiagramMiniPauli. ... Image File history File links GroupDiagramMiniG44. ...

Small 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: A computer algebra system (CAS) is a software program that facilitates symbolic mathematics. ... GAP (Groups, Algorithms and Programming) is a computer algebra system for computational discrete algebra similar to Mathematica with particular emphasis on, but not restricted to, computational group theory. ... In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations. ...

• those of order at most 2000 except for order 1024 (423 164 062 groups);
• those of order 5⁵ and 7⁴ (92 groups);
• those of order qⁿ×p where qⁿ 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 .

See also

• Small Latin squares and quasigroups

Below the Latin squares and quasigroups of some small orders are considered. ...

External links

• Small groups