List of small groups Totally Explained

Everything about List Of Small Groups totally explained

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/nZ).
Dih_n: the dihedral group of order 2n (often the notation D_n or D_2n 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 4n.

The notations Z_n and Dih_n have the advantage that point groups in three dimensions C_n and D_n don't 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; Gⁿ 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's 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 doesn't 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're cyclic groups, or direct products thereof; see abelian groups.

Order	Group	Subgroups	Properties
1	trivial group = Z₁ = S₁ = A₂	-	various properties hold trivially
2	Z₂ = S₂ = Dih₁	-	simple, the smallest non-trivial group
3	Z₃ = A₃	-	simple
4	Z₄	Z₂
4	Klein four-group = Z₂ × Z₂ = Dih₂	Z₂ (3)	the smallest non-cyclic group
5	Z₅	-	simple
6	Z₆ = Z₃ × Z₂	Z₃, Z₂
7	Z₇	-	simple
8	Z₈	Z₄, Z₂
	Z₄ × Z₂	Z₂², Z₄ (2), Z₂ (3)
	Z₂³	Z₂² (7), Z₂ (7)	the non-identity elements correspond to the points in the Fano plane, the Z₂ × Z₂ subgroups to the lines
9	Z₉	Z₃
9	Z₃²	Z₃ (4)
10	Z₁₀ = Z₅ × Z₂	Z₅, Z₂
11	Z₁₁	-	simple
12	Z₁₂ = Z₄ × Z₃	Z₆, Z₄, Z₃, Z₂
12	Z₆ × Z₂ = Z₃ × Z₂²	Z₆ (3), Z₃, Z₂ (3), Z₂²
13	Z₁₃	-	simple
14	Z₁₄ = Z₇ × Z₂	Z₇, Z₂
15	Z₁₅ = Z₅ × Z₃	Z₅, Z₃
16	Z₁₆	Z₈, Z₄, Z₂
	Z₂⁴	Z₂ (15), Z₂² (35), Z₂³ (15)
	Z₄ × Z₂²	Z₂ (7), Z₄ (4), Z₂² (7), Z₂³, Z₄ × Z₂ (6)
	Z₈ × Z₂	Z₂ (3), Z₄ (2), Z₂², Z₈ (2), Z₄ × Z₂
	Z₄²	Z₂ (3), Z₄ (6), Z₂², Z₄ × Z₂ (3)

List of small non-abelian groups

Order	Group	Subgroups	Properties
6	S₃ = Dih₃	Z₃, Z₂ (3)	the smallest non-abelian group
8	Dih₄	Z₄, Z₂² (2), Z₂ (5)
8	Quaternion group, Q₈ = Dic₂	Z₄ (3), Z₂	the smallest Hamiltonian group
10	Dih₅	Z₅, Z₂ (5)
12	Dih₆ = Dih₃ × Z₂	Z₆, Dih₃ (2), Z₂² (3), Z₃, Z₂ (7)
	A₄	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 ⋊ Z₄	Z₂, Z₃, Z₄ (3), Z₆
14	Dih₇	Z₇, Z₂ (7)
16	Dih₈	Z₈, Dih₄ (2), Z₂² (4), Z₄, Z₂ (9)
	Dih₄ × Z₂	Dih₄ (2), Z₄ × Z₂, Z₂³ (2), Z₂² (7), Z₄ (2), Z₂ (11)
	Generalized quaternion group, Q₁₆ = Dic₄
	Q₈ × Z₂		Hamiltonian
	The order 16 quasidihedral group
	The order 16 modular group
	Z₄ ⋊ Z₄
	The group generated by the Pauli matrices
	G_4,4 = Z₂² ⋊ Z₄

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:

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 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 .

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