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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, ... (sequence A000001 in the OEIS)

For labeled groups, see OEIS: A034383.

Glossary

Each group is named by Small Groups library as Go, where o is the order of the group, and i is the index used to label the group within that order.

Common group names:

The notations Zn and Dihn have the advantage that point groups in three dimensions Cn and Dn do not have the same notation. There are more isometry groups than these two, of the same abstract group type.

The notation G × H denotes the direct product of the two groups; G 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 n < 60, the simple groups are precisely the cyclic groups Zn, 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 <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, ... (sequence A000688 in the OEIS)

For labeled abelian groups, see OEIS: A034382.

List of all abelian groups up to order 31
Order Id. Go Group Non-trivial proper subgroups Cycle
graph 		Properties
1 1 G1 Z1 = S1 = A2 – Trivial. Cyclic. Alternating. Symmetric. Elementary.
2 2 G2 Z2 = S2 = D2 – Simple. Symmetric. Cyclic. Elementary. (Smallest non-trivial group.)
3 3 G3 Z3 = A3 – Simple. Alternating. Cyclic. Elementary.
4 4 G4 Z4 = Dic1 Z2 Cyclic.
5 G4 Z2 = K4 = D4 Z2 (3) Elementary. Product. (Klein four-group. The smallest non-cyclic group.)
5 6 G5 Z5 – Simple. Cyclic. Elementary.
6 8 G6 Z6 = Z3 × Z2 Z3, Z2 Cyclic. Product.
7 9 G7 Z7 – Simple. Cyclic. Elementary.
8 10 G8 Z8 Z4, Z2 Cyclic.
11 G8 Z4 × Z2 Z2, Z4 (2), Z2 (3) Product.
14 G8 Z2 Z2 (7), Z2 (7) Product. Elementary. (The non-identity elements correspond to the points in the Fano plane, the Z2 × Z2 subgroups to the lines.)
9 15 G9 Z9 Z3 Cyclic.
16 G9 Z3 Z3 (4) Elementary. Product.
10 18 G10 Z10 = Z5 × Z2 Z5, Z2 Cyclic. Product.
11 19 G11 Z11 – Simple. Cyclic. Elementary.
12 21 G12 Z12 = Z4 × Z3 Z6, Z4, Z3, Z2 Cyclic. Product.
24 G12 Z6 × Z2 = Z3 × Z2 Z6 (3), Z3, Z2 (3), Z2 Product.
13 25 G13 Z13 – Simple. Cyclic. Elementary.
14 27 G14 Z14 = Z7 × Z2 Z7, Z2 Cyclic. Product.
15 28 G15 Z15 = Z5 × Z3 Z5, Z3 Cyclic. Product.
16 29 G16 Z16 Z8, Z4, Z2 Cyclic.
30 G16 Z4 Z2 (3), Z4 (6), Z2, Z4 × Z2 (3) Product.
33 G16 Z8 × Z2 Z2 (3), Z4 (2), Z2, Z8 (2), Z4 × Z2 Product.
38 G16 Z4 × Z2 Z2 (7), Z4 (4), Z2 (7), Z2, Z4 × Z2 (6) Product.
42 G16 Z2 = K4 Z2 (15), Z2 (35), Z2 (15) Product. Elementary.
17 43 G17 Z17 – Simple. Cyclic. Elementary.
18 45 G18 Z18 = Z9 × Z2 Z9, Z6, Z3, Z2 Cyclic. Product.
48 G18 Z6 × Z3 = Z3 × Z2 Z2, Z3 (4), Z6 (4), Z3 Product.
19 49 G19 Z19 – Simple. Cyclic. Elementary.
20 51 G20 Z20 = Z5 × Z4 Z10, Z5, Z4, Z2 Cyclic. Product.
54 G20 Z10 × Z2 = Z5 × Z2 Z2 (3), K4, Z5, Z10 (3) Product.
21 56 G21 Z21 = Z7 × Z3 Z7, Z3 Cyclic. Product.
22 58 G22 Z22 = Z11 × Z2 Z11, Z2 Cyclic. Product.
23 59 G23 Z23 – Simple. Cyclic. Elementary.
24 61 G24 Z24 = Z8 × Z3 Z12, Z8, Z6, Z4, Z3, Z2 Cyclic. Product.
68 G24 Z12 × Z2 = Z6 × Z4 =
Z4 × Z3 × Z2 		Z12, Z6, Z4, Z3, Z2 Product.
74 G24 Z6 × Z2 = Z3 × Z2 Z6, Z3, Z2 Product.
25 75 G25 Z25 Z5 Cyclic.
76 G25 Z5 Z5 (6) Product. Elementary.
26 78 G26 Z26 = Z13 × Z2 Z13, Z2 Cyclic. Product.
27 79 G27 Z27 Z9, Z3 Cyclic.
80 G27 Z9 × Z3 Z9, Z3 Product.
83 G27 Z3 Z3 Product. Elementary.
28 85 G28 Z28 = Z7 × Z4 Z14, Z7, Z4, Z2 Cyclic. Product.
87 G28 Z14 × Z2 = Z7 × Z2 Z14, Z7, Z4, Z2 Product.
29 88 G29 Z29 – Simple. Cyclic. Elementary.
30 92 G30 Z30 = Z15 × Z2 = Z10 × Z3 =
Z6 × Z5 = Z5 × Z3 × Z2 		Z15, Z10, Z6, Z5, Z3, Z2 Cyclic. Product.
31 93 G31 Z31 – Simple. Cyclic. Elementary.

List of small non-abelian groups

The numbers of non-abelian groups, by order, are counted by (sequence A060689 in the OEIS). 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, ... (sequence A060652 in the OEIS)
List of all nonabelian groups up to order 31
Order Id. Go Group Non-trivial proper subgroups Cycle
graph 		Properties
6 7 G6 D6 = S3 = Z3 ⋊ Z2 Z3, Z2 (3) Dihedral group, Dih3, the smallest non-abelian group, symmetric group, smallest Frobenius group.
8 12 G8 D8 Z4, Z2 (2), Z2 (5) Dihedral group, Dih4. Extraspecial group. Nilpotent.
13 G8 Q8 Z4 (3), Z2 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 G10 D10 Z5, Z2 (5) Dihedral group, Dih5, Frobenius group.
12 20 G12 Q12 = Z3 ⋊ Z4 Z2, Z3, Z4 (3), Z6 Dicyclic group Dic3, Binary dihedral group, <3,2,2>
22 G12 A4 = K4 ⋊ Z3 = (Z2 × Z2) ⋊ Z3 Z2, Z3 (4), Z2 (3) Alternating group. No subgroups of order 6, although 6 divides its order. Smallest Frobenius group that is not a dihedral group.
Chiral tetrahedral symmetry (T)
23 G12 D12 = D6 × Z2 Z6, D6 (2), Z2 (3), Z3, Z2 (7) Dihedral group, Dih6, product.
14 26 G14 D14 Z7, Z2 (7) Dihedral group, Dih7, Frobenius group
16 31 G16 G4,4 = K4 ⋊ Z4 Z2, Z4 × Z2 (2), Z4 (4), Z2 (7), Z2 (7) Has the same number of elements of every order as the Pauli group. Nilpotent.
32 G16 Z4 ⋊ Z4 Z2 × Z2 (3), Z4 (6), Z2, Z2 (3) The squares of elements do not form a subgroup. Has the same number of elements of every order as Q8 × Z2. Nilpotent.
34 G16 Z8 ⋊ Z2 Z8 (2), Z2 × Z2, Z4 (2), Z2, Z2 (3) Sometimes called the modular group of order 16, though this is misleading as abelian groups and Q8 × Z2 are also modular. Nilpotent.
35 G16 D16 Z8, D8 (2), Z2 (4), Z4, Z2 (9) Dihedral group, Dih8. Nilpotent.
36 G16 QD16 Z8, Q8, D8, Z4 (3), Z2 (2), Z2 (5) The order 16 quasidihedral group. Nilpotent.
37 G16 Q16 Z8, Q8 (2), Z4 (5), Z2 Generalized quaternion group, Dicyclic group Dic4, binary dihedral group, <4,2,2>. Nilpotent.
39 G16 D8 × Z2 D8 (4), Z4 × Z2, Z2 (2), Z2 (13), Z4 (2), Z2 (11) Product. Nilpotent.
40 G16 Q8 × Z2 Q8 (4), Z2 × Z2 (3), Z4 (6), Z2, Z2 (3) Hamiltonian group, product. Nilpotent.
41 G16 (Z4 × Z2) ⋊ Z2 Q8, D8 (3), Z4 × Z2 (3), Z4 (4), Z2 (3), Z2 (7) The Pauli group generated by the Pauli matrices. Nilpotent.
18 44 G18 D18 Z9, D6 (3), Z3, Z2 (9) Dihedral group, Dih9, Frobenius group.
46 G18 Z3 ⋊ Z6 = D6 × Z3 = S3 × Z3 Z3, D6, Z6 (3), Z3 (4), Z2 (3) Product.
47 G18 (Z3 × Z3) ⋊ Z2 Z3, D6 (12), Z3 (4), Z2 (9) Frobenius group.
20 50 G20 Q20 Z10, Z5, Z4 (5), Z2 Dicyclic group Dic5, Binary dihedral group, <5,2,2>.
52 G20 Z5 ⋊ Z4 D10, Z5, Z4 (5), Z2 (5) Frobenius group.
53 G20 D20 = D10 × Z2 Z10, D10 (2), Z5, Z2 (5), Z2 (11) Dihedral group, Dih10, product.
21 55 G21 Z7 ⋊ Z3 Z7, Z3 (7) Smallest non-abelian group of odd order. Frobenius group.
22 57 G22 D22 Z11, Z2 (11) Dihedral group Dih11, Frobenius group.
24 60 G24 Z3 ⋊ Z8 Z12, Z8 (3), Z6, Z4, Z3, Z2 Central extension of S3.
62 G24 SL(2,3) = Q8 ⋊ Z3 Q8, Z6 (4), Z4 (3), Z3 (4), Z2 Binary tetrahedral group, 2T = <3,3,2>.
63 G24 Q24 = Z3 ⋊ Q8 Z12, Q12 (2), Q8 (3), Z6, Z4 (7), Z3, Z2 Dicyclic group Dic6, Binary dihedral, <6,2,2>.
64 G24 D6 × Z4 = S3 × Z4 Z12, D12, Q12, Z4 × Z2 (3), Z6, D6 (2), Z4 (4), Z2 (3), Z3, Z2 (7) Product.
65 G24 D24 Z12, D12 (2), D8 (3), Z6, D6 (4), Z4, Z2 (6), Z3, Z2 (13) Dihedral group, Dih12.
66 G24 Q12 × Z2 = Z2 × (Z3 ⋊ Z4) Z6 × Z2, Q12 (2), Z4 × Z2 (3), Z6 (3), Z4 (6), Z2, Z3, Z2 (3) Product.
67 G24 (Z6 × Z2) ⋊ Z2 = Z3 ⋊ Dih4 Z6 × Z2, D12, Q12, D8 (3), Z6 (3), D6 (2), Z4 (3), Z2 (4), Z3, Z2 (9) Double cover of dihedral group.
69 G24 D8 × Z3 Z12, Z6 × Z2 (2), D8, Z6 (5), Z4, Z2 (2), Z3, Z2 (5) Product. Nilpotent.
70 G24 Q8 × Z3 Z12 (3), Q8, Z6, Z4 (3), Z3, Z2 Product. Nilpotent.
71 G24 S4 A4, D8 (3), D6 (4), Z4 (3), Z2 (4), Z3 (4), Z2 (9) Symmetric group. Has no normal Sylow subgroups. Chiral octahedral symmetry (O), Achiral tetrahedral symmetry (Td)
72 G24 A4 × Z2 A4, Z2, Z6 (4), Z2 (7), Z3 (4), Z2 (7) Product. Pyritohedral symmetry (Th)
73 G24 D12 × Z2 Z6 × Z2, D12 (6), Z2 (3), Z6 (3), D6 (4), Z2 (19), Z3, Z2 (15) Product.
26 77 G26 D26 Z13, Z2 (13) Dihedral group, Dih13, Frobenius group.
27 81 G27 Z3 ⋊ Z3 Z3 (4), Z3 (13) All non-trivial elements have order 3. Extraspecial group. Nilpotent.
82 G27 Z9 ⋊ Z3 Z9 (3), Z3, Z3 (4) Extraspecial group. Nilpotent.
28 84 G28 Z7 ⋊ Z4 Z14, Z7, Z4 (7), Z2 Dicyclic group Dic7, Binary dihedral group, <7,2,2>.
86 G28 D28 = D14 × Z2 Z14, D14 (2), Z7, Z2 (7), Z2 (9) Dihedral group, Dih14, product.
30 89 G30 D6 × Z5 Z15, Z10 (3), D6, Z5, Z3, Z2 (3) Product.
90 G30 D10 × Z3 Z15, D10, Z6 (5), Z5, Z3, Z2 (5) Product.
91 G30 D30 Z15, D10 (3), D6 (5), Z5, Z3, Z2 (15) Dihedral group, Dih15, Frobenius group.

Classifying groups of small order

Small groups of prime power order p are given as follows:

  • Order p: The only group is cyclic.
  • Order p: There are just two groups, both abelian.
  • Order p: 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 by a cyclic group of order p. The other is the quaternion group for p = 2 and a group of exponent p for p > 2.
  • Order p: 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 S4 × Z2
  • Order 60: The alternating group A5.

The smallest order for which it is not known how many nonisomorphic groups there are is 2048 = 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 (423164062 groups in the library; the ones of order 1024 had to be skipped, as there are additional 49487367289 nonisomorphic 2-groups of order 1024);
  • those of cubefree order at most 50000 (395 703 groups);
  • those of squarefree order;
  • those of order p for n at most 6 and p prime;
  • those of order p for p = 3, 5, 7, 11 (907 489 groups);
  • those of order pq where q divides 2, 3, 5 or 7 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

Notes

  1. ^ Identifier when groups are numbered by order, o, then by index, i, from the small groups library, starting at 1.
  1. ^ Dockchitser, Tim. "Group Names". Retrieved 23 May 2023.
  2. See a worked example showing the isomorphism Z6 = Z3 × Z2.
  3. Chen, Jing; Tang, Lang (2020). "The Commuting Graphs on Dicyclic Groups". Algebra Colloquium. 27 (4): 799–806. doi:10.1142/S1005386720000668. ISSN 1005-3867. S2CID 228827501.
  4. ^ Coxeter, H. S. M. (1957). Generators and relations for discrete groups. Berlin: Springer. doi:10.1007/978-3-662-25739-5. ISBN 978-3-662-23654-3. <l,m,n>: R=S=T=RST:
  5. Wild, Marcel (2005). "The Groups of Order Sixteen Made Easy" (PDF). Am. Math. Mon. 112 (1): 20–31. doi:10.1080/00029890.2005.11920164. JSTOR 30037381. S2CID 15362871. Archived from the original (PDF) on 2006-09-23.
  6. "Subgroup structure of symmetric group:S4 - Groupprops".
  7. Eick, Bettina; Horn, Max; Hulpke, Alexander (2018). Constructing groups of Small Order: Recent results and open problems (PDF). Springer. pp. 199–211. doi:10.1007/978-3-319-70566-8_8. ISBN 978-3-319-70566-8.
  8. Hans Ulrich Besche The Small Groups library Archived 2012-03-05 at the Wayback Machine
  9. "Numbers of isomorphism types of finite groups of given order". www.icm.tu-bs.de. Archived from the original on 2019-07-25. Retrieved 2017-04-05.
  10. Burrell, David (2021-12-08). "On the number of groups of order 1024". Communications in Algebra. 50 (6): 2408–2410. doi:10.1080/00927872.2021.2006680.

References

  • Coxeter, H. S. M. & Moser, W. O. J. (1980). Generators and Relations for Discrete Groups. New York: Springer-Verlag. ISBN 0-387-09212-9., Table 1, Nonabelian groups order<32.
  • Hall, Jr., Marshall; Senior, James K. (1964). "The Groups of Order 2 (n ≤ 6)". MathSciNet. Macmillan. MR 0168631. A catalog of the 340 groups of order dividing 64 with tables of defining relations, constants, and lattice of subgroups of each group.

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