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 Small Groups Library as G_oⁱ, where o is the order of the group, and i is the index used to label 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)
• D_n: the dihedral group of order 2n, the same as Dih_n (notation used in section List of small non-abelian groups)
• K₄: the Klein four-group of order 4, isomorphic to 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_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 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 , the simple groups are precisely the cyclic groups Z_n, for prime n.)

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 <relations> 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 .

{| class="wikitable skin-invert-image"

|+ List of all abelian groups up to order 31

|-

! Order

! Id.

! G_oⁱ

! Group

! Non-trivial proper subgroups

! Cycle<br />graph

! Properties

|-

! 1

! 1

! G₁¹

| Z₁ ≅ S₁ ≅ A₂

| –

| align=center|40px

| Trivial. Cyclic. Alternating. Symmetric. Elementary.

|-

! 2

! 2

! G₂¹

| Z₂ ≅ S₂ ≅ D₁

| –

| align=center|40px

| Simple. Symmetric. Cyclic. Elementary. (Smallest non-trivial group.)

|-

! 3

! 3

! G₃¹

| Z₃ ≅ A₃

| –

| align=center|40px

| Simple. Alternating. Cyclic. Elementary.

|-

! rowspan="2" | 4

! 4

! G₄¹

| Z₄ ≅ Q₄

| Z₂

| align=center|40px

| Cyclic.

|-

! 5

! G₄²

| Z₂² ≅ K₄ ≅ D₂

| Z₂ (3)

| align=center|40px

| Elementary. Product. (Klein four-group. The smallest non-cyclic group.)

|-

! 5

! 6

! G₅¹

| Z₅

| –

| align=center|40px

| Simple. Cyclic. Elementary.

|-

! 6

! 8

! G₆²

| Z₆ ≅ Z₃ × Z₂

| Z₃, Z₂

| align=center|40px

| Cyclic. Product.

|-

! 7

! 9

! G₇¹

| Z₇

| –

| align=center|40px

| Simple. Cyclic. Elementary.

|-

! rowspan="3" | 8

! 10

! G<sub name=g8>8</sub>¹

| Z₈

| Z₄, Z₂

| align=center|40px

| Cyclic.

|-

! 11

! G₈²

| Z₄ × Z₂

| Z₂², Z₄ (2), Z₂ (3)

| align=center|40px

| Product.

|-

! 14

! G₈⁵

| Z₂³

| Z₂² (7), Z₂ (7)

| align=center|40px

| Product. Elementary. (The non-identity elements correspond to the points in the Fano plane, the subgroups to the lines.)

|-

! rowspan="2" | 9

! 15

! G₉¹

| Z₉

| Z₃

| align=center|40px

| Cyclic.

|-

! 16

! G₉²

| Z₃²

| Z₃ (4)

|align=center| 40px

| Elementary. Product.

|-

! 10

! 18

! G₁₀²

| Z₁₀ ≅ Z₅ × Z₂

| Z₅, Z₂

| align=center|40px

| Cyclic. Product.

|-

! 11

! 19

! G₁₁¹

| Z₁₁

| –

| align=center|40px

| Simple. Cyclic. Elementary.

|-

! rowspan="2" | 12

! 21

! G₁₂²

| Z₁₂ ≅ Z₄ × Z₃

| Z₆, Z₄, Z₃, Z₂

| align=center|40px

| Cyclic. Product.

|-

! 24

! G₁₂⁵

| Z₆ × Z₂ ≅ Z₃ × Z₂²

| Z₆ (3), Z₃, Z₂ (3), Z₂²

| align=center|40px

| Product.

|-

! 13

! 25

! G₁₃¹

| Z₁₃

| –

| align=center|40px

| Simple. Cyclic. Elementary.

|-

! 14

! 27

! G₁₄²

| Z₁₄ ≅ Z₇ × Z₂

| Z₇, Z₂

|align=center| 40px

| Cyclic. Product.

|-

! 15

! 28

! G₁₅¹

| Z₁₅ ≅ Z₅ × Z₃

| Z₅, Z₃

| align=center|40px

| Cyclic. Product.

|-

! rowspan="5" | 16

! 29

! G₁₆¹

| Z₁₆

| Z₈, Z₄, Z₂

| align=center|40px

| Cyclic.

|-

! 30

! G₁₆²

| Z₄²

| Z₂ (3), Z₄ (6), Z₂², (3)</td>

| align=center|40px

| Product.

|-

! 33

! G₁₆⁵

| Z₈ × Z₂

| Z₂ (3), Z₄ (2), Z₂², Z₈ (2),

| align=center|40px

| Product.

|-

! 38

! G₁₆¹⁰

| Z₄ × Z₂²

| Z₂ (7), Z₄ (4), Z₂² (7), Z₂³, (6)

| align=center|40px

| Product.

|-

! 42

! G₁₆¹⁴

| Z₂⁴ ≅ K₄²

| Z₂ (15), Z₂² (35), Z₂³ (15)</td>

| align=center|40px

| Product. Elementary.

|-

! 17

! 43

! G₁₇¹

| Z₁₇

| –

| align=center|40px

| Simple. Cyclic. Elementary.

|-

! rowspan="2" | 18

! 45

! G₁₈²

| Z₁₈ ≅ Z₉ × Z₂

| Z₉, Z₆, Z₃, Z₂

| align=center|40px

| Cyclic. Product.

|-

! 48

! G₁₈⁵

| Z₆ × Z₃ ≅ Z₃² × Z₂ || Z₂, Z₃ (4), Z₆ (4), Z₃² ||50px || Product.

|-

! 19

! 49

! G₁₉¹

| Z₁₉

| –

| align=center|40px

| Simple. Cyclic. Elementary.

|-

! rowspan="2" | 20

! 51

! G₂₀²

| Z₂₀ ≅ Z₅ × Z₄

| Z₁₀, Z₅, Z₄, Z₂

| align=center|40px

| Cyclic. Product.

|-

! 54

! G₂₀⁵

| Z₁₀ × Z₂ ≅ Z₅ × Z₂² ||Z₂ (3), K₄, Z₅, Z₁₀ (3)

| align=center|40px

| Product.

|-

! 21

! 56

! G₂₁²

| Z₂₁ ≅ Z₇ × Z₃

| Z₇, Z₃

| align=center|40px

| Cyclic. Product.

|-

! 22

! 58

! G₂₂²

| Z₂₂ ≅ Z₁₁ × Z₂

| Z₁₁, Z₂

| align=center|40px

| Cyclic. Product.

|-

! 23

! 59

! G₂₃¹

| Z₂₃

| –

| align=center|40px

| Simple. Cyclic. Elementary.

|-

! rowspan=3|24

! 61

! G₂₄²

| Z₂₄ ≅ Z₈ × Z₃

| Z₁₂, Z₈, Z₆, Z₄, Z₃, Z₂

| align=center|40px

| Cyclic. Product.

|-

! 68

! G₂₄⁹

| Z₁₂ × Z₂ ≅ Z₆ × Z₄ ≅ <br />Z₄ × Z₃ × Z₂

| Z₁₂, Z₆, Z₄, Z₃, Z₂

|

| Product.

|-

! 74

! G₂₄¹⁵

| Z₆ × Z₂² ≅ Z₃ × Z₂³

| Z₆, Z₃, Z₂

|

| Product.

|-

! rowspan=2|25

! 75

! G₂₅¹

| Z₂₅

| Z₅

|

| Cyclic.

|-

! 76

! G₂₅²

| Z₅²

| Z₅ (6)

|

| Product. Elementary.

|-

! 26

! 78

! G₂₆²

| Z₂₆ ≅ Z₁₃ × Z₂

| Z₁₃, Z₂

|

| Cyclic. Product.

|-

! rowspan=3|27

! 79

! G₂₇¹

| Z₂₇ ||Z₉, Z₃

|

| Cyclic.

|-

! 80

! G₂₇²

| Z₉ × Z₃

| Z₉, Z₃

|

| Product.

|-

! 83

! G₂₇⁵

| Z₃³ || Z₃ || || Product. Elementary.

|-

! rowspan=2|28

! 85

! G₂₈²

| Z₂₈ ≅ Z₇ × Z₄ || Z₁₄, Z₇, Z₄, Z₂ || || Cyclic. Product.

|-

! 87

! G₂₈⁴

| Z₁₄ × Z₂ ≅ Z₇ × Z₂² || Z₁₄, Z₇, Z₄, Z₂

|

| Product.

|-

! 29

! 88

! G₂₉¹

| Z₂₉

| –

|

| Simple. Cyclic. Elementary.

|-

! 30

! 92

! G₃₀⁴

| style="white-space:nowrap;" | Z₃₀ ≅ Z₁₅ × Z₂ ≅ Z₁₀ × Z₃ ≅ <br />Z₆ × Z₅ ≅ Z₅ × Z₃ × Z₂

| Z₁₅, Z₁₀, Z₆, Z₅, Z₃, Z₂

|

| Cyclic. Product.

|-

! 31

! 93

! G₃₁¹

| Z₃₁

| –

|

| Simple. Cyclic. Elementary.

|}

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, ...

{| class="wikitable skin-invert-image"

|+ List of all nonabelian groups up to order 31

|-

! Order

! Id.

! G_oⁱ

! Group

! Non-trivial proper subgroups

! Cycle <br />graph

! Properties

|-

! 6

! 7

! G₆¹

| D₃ ≅ S₃ ≅ Z₃ <math>\rtimes</math> Z₂

| Z₃, Z₂ (3)

| 40px

| Dihedral group, Dih₃, the smallest non-abelian group, symmetric group, smallest Frobenius group.

|-

! rowspan=2 | 8

! 12

! G₈³

| D₄ ≅ Z₄ <math>\rtimes</math> Z₂

| Z₄, Z₂² (2), Z₂ (5)

| 40px

| Dihedral group, Dih₄. Extraspecial group. Nilpotent.

|-

! 13

! G₈⁴

| Q₈ ≅ Z₄ <math>.</math> Z₂

| Z₄ (3), Z₂

| 40px

| Quaternion group, Hamiltonian group (all subgroups are normal without the group being abelian). The smallest group G demonstrating that for a normal subgroup H the quotient group G/H need not be isomorphic to a subgroup of G. Extraspecial group. Dic₂, Binary dihedral group <2,2,2>. Nilpotent.

|-

! 10

! 17

! G₁₀¹

| D₅ ≅ Z₅ <math>\rtimes</math> Z₂

| Z₅, Z₂ (5)

| 40px

| Dihedral group, Dih₅, Frobenius group.

|-

! rowspan=3 | 12

! 20

! G₁₂¹

| style="white-space:nowrap;" | Q₁₂ ≅ Z₃ ⋊ Z₄

| Z₂, Z₃, Z₄ (3), Z₆

| 40px

| Dicyclic group Dic₃, Binary dihedral group, <3,2,2>.

! 31

! G₁₆³

| K₄ ⋊ Z₄

| Z₂³, Z₄ × Z₂ (2), Z₄ (4), Z₂² (7), Z₂ (7)

| 40px

| Has the same number of elements of every order as the Pauli group. Nilpotent.

|-

! 32

! G₁₆⁴

| Z₄ ⋊ Z₄

| Z₄ × Z₂ (3), Z₄ (6), Z₂², Z₂ (3)

| 40px

| The squares of elements do not form a subgroup. Has the same number of elements of every order as Q₈ × Z₂. Nilpotent.

|-

! 34

! G₁₆⁶

| Z₈ ⋊₅ Z₂

| Z₈ (2), Z₄ × Z₂, Z₄ (2), Z₂², Z₂ (3)

| 40px

| Sometimes called the modular group of order 16, though this is misleading as abelian groups and Q₈ × Z₂ are also modular. Nilpotent.

|-

! 35

! G₁₆⁷

| D₈ ≅ Z₈ ⋊₋₁ Z₂

| Z₈, D₄ (2), Z₂² (4), Z₄, Z₂ (9)

| 40px

| Dihedral group, Dih₈. Nilpotent.

|-

! 36

! G₁₆⁸

| QD₁₆ ≅ Z₈ ⋊₃ Z₂

| Z₈, Q₈, D₄, Z₄ (3), Z₂² (2), Z₂ (5)

| 40px

| The order 16 quasidihedral group. Nilpotent.

|-

! 37

! G₁₆⁹

| Q₁₆

| Z₈, Q₈ (2), Z₄ (5), Z₂

| 40px

| Generalized quaternion group, Dicyclic group Dic₄, binary dihedral group, <4,2,2>.

| 40px || Symmetric group. Has no normal Sylow subgroups. Chiral octahedral symmetry (O), Achiral tetrahedral symmetry (T_d).

|-

! 72

! G₂₄¹³

| A₄ × Z₂

| A₄, Z₂³, Z₆ (4), Z₂² (7), Z₃ (4), Z₂ (7)

| 40px || Product. Pyritohedral symmetry (T_h).

|-

! 73

! G₂₄¹⁴

| D₆ × Z₂

| Z₆ × Z₂, D₆ (6), Z₂³ (3), Z₆ (3), D₃ (4), Z₂² (19), Z₃, Z₂ (15)

| || Product.

|-

! 26

! 77

! G₂₆¹

| D₁₃ ≅ Z₁₃ <math>\rtimes</math> Z₂

| Z₁₃, Z₂ (13)

| || Dihedral group, Dih₁₃, Frobenius group.

|-

! rowspan=2 | 27

! 81

! G₂₇³

| Z₃² ⋊ Z₃

| Z₃² (4), Z₃ (13)

| || All non-trivial elements have order 3. Extraspecial group. Nilpotent.

|-

! 82

! G₂₇⁴

| Z₉ ⋊ Z₃

| Z₉ (3), Z₃², Z₃ (4)

| || Extraspecial group. Nilpotent.

|-

! rowspan=2 | 28

! 84

! G₂₈¹

| Z₇ ⋊ Z₄

| Z₁₄, Z₇, Z₄ (7), Z₂

| || Dicyclic group Dic₇, Binary dihedral group, <7,2,2>.

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:

• 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 additional 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 (not necessarily distinct).

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

The smallest order for which the Small Groups 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.
• A catalog of the 340 groups of order dividing 64 with tables of defining relations, constants, and lattice of subgroups of each group.

External links

• Particular groups in the Group Properties Wiki
• GroupNames database
• Hall, Jr., Marshall; Senior, James Kuhn (1964). The Groups of Order 2ⁿ (n ≤ 6). New York: Macmillan / London: Collier-Macmillan Ltd. LCCN 64016861