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List of small groups

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The following list in mathematics contains the finite groups of small order up to group isomorphism.

Counts

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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

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Each group is named by Small Groups library as Goi, 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; 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 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

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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.[a] Goi Group Non-trivial proper subgroups[1] Cycle
graph 		Properties
1 1 G11 Z1 = S1 = A2 – Trivial. Cyclic. Alternating. Symmetric. Elementary.
2 2 G21 Z2 = S2 = D2 – Simple. Symmetric. Cyclic. Elementary. (Smallest non-trivial group.)
3 3 G31 Z3 = A3 – Simple. Alternating. Cyclic. Elementary.
4 4 G41 Z4 = Dic1 Z2 Cyclic.
5 G42 Z22 = K4 = D4 Z2 (3) Elementary. Product. (Klein four-group. The smallest non-cyclic group.)
5 6 G51 Z5 – Simple. Cyclic. Elementary.
6 8 G62 Z6 = Z3 × Z2[2] Z3, Z2 Cyclic. Product.
7 9 G71 Z7 – Simple. Cyclic. Elementary.
8 10 G81 Z8 Z4, Z2 Cyclic.
11 G82 Z4 × Z2 Z22, Z4 (2), Z2 (3) Product.
14 G85 Z23 Z22 (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 G91 Z9 Z3 Cyclic.
16 G92 Z32 Z3 (4) Elementary. Product.
10 18 G102 Z10 = Z5 × Z2 Z5, Z2 Cyclic. Product.
11 19 G111 Z11 – Simple. Cyclic. Elementary.
12 21 G122 Z12 = Z4 × Z3 Z6, Z4, Z3, Z2 Cyclic. Product.
24 G125 Z6 × Z2 = Z3 × Z22 Z6 (3), Z3, Z2 (3), Z22 Product.
13 25 G131 Z13 – Simple. Cyclic. Elementary.
14 27 G142 Z14 = Z7 × Z2 Z7, Z2 Cyclic. Product.
15 28 G151 Z15 = Z5 × Z3 Z5, Z3 Cyclic. Product.
16 29 G161 Z16 Z8, Z4, Z2 Cyclic.
30 G162 Z42 Z2 (3), Z4 (6), Z22, Z4 × Z2 (3) Product.
33 G165 Z8 × Z2 Z2 (3), Z4 (2), Z22, Z8 (2), Z4 × Z2 Product.
38 G1610 Z4 × Z22 Z2 (7), Z4 (4), Z22 (7), Z23, Z4 × Z2 (6) Product.
42 G1614 Z24 = K42 Z2 (15), Z22 (35), Z23 (15) Product. Elementary.
17 43 G171 Z17 – Simple. Cyclic. Elementary.
18 45 G182 Z18 = Z9 × Z2 Z9, Z6, Z3, Z2 Cyclic. Product.
48 G185 Z6 × Z3 = Z32 × Z2 Z2, Z3 (4), Z6 (4), Z32 Product.
19 49 G191 Z19 – Simple. Cyclic. Elementary.
20 51 G202 Z20 = Z5 × Z4 Z10, Z5, Z4, Z2 Cyclic. Product.
54 G205 Z10 × Z2 = Z5 × Z22 Z2 (3), K4, Z5, Z10 (3) Product.
21 56 G212 Z21 = Z7 × Z3 Z7, Z3 Cyclic. Product.
22 58 G222 Z22 = Z11 × Z2 Z11, Z2 Cyclic. Product.
23 59 G231 Z23 – Simple. Cyclic. Elementary.
24 61 G242 Z24 = Z8 × Z3 Z12, Z8, Z6, Z4, Z3, Z2 Cyclic. Product.
68 G249 Z12 × Z2 = Z6 × Z4 =
Z4 × Z3 × Z2 		Z12, Z6, Z4, Z3, Z2 Product.
74 G2415 Z6 × Z22 = Z3 × Z23 Z6, Z3, Z2 Product.
25 75 G251 Z25 Z5 Cyclic.
76 G252 Z52 Z5 (6) Product. Elementary.
26 78 G262 Z26 = Z13 × Z2 Z13, Z2 Cyclic. Product.
27 79 G271 Z27 Z9, Z3 Cyclic.
80 G272 Z9 × Z3 Z9, Z3 Product.
83 G275 Z33 Z3 Product. Elementary.
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 Z30 = Z15 × Z2 = Z10 × Z3 =
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

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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.[a] Goi Group Non-trivial proper subgroups[1] Cycle
graph 		Properties
6 7 G61 D6 = S3 = Z3 ⋊ Z2 Z3, Z2 (3) Dihedral group, Dih3, the smallest non-abelian group, symmetric group, smallest Frobenius group.
8 12 G83 D8 Z4, Z22 (2), Z2 (5) Dihedral group, Dih4. Extraspecial group. Nilpotent.
13 G84 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,[3] Binary dihedral group <2,2,2>.[4] Nilpotent.
10 17 G101 D10 Z5, Z2 (5) Dihedral group, Dih5, Frobenius group.
12 20 G121 Q12 = Z3 ⋊ Z4 Z2, Z3, Z4 (3), Z6 Dicyclic group Dic3, Binary dihedral group, <3,2,2>[4]
22 G123 A4 = K4 ⋊ Z3 = (Z2 × Z2) ⋊ Z3 Z22, 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 G124 D12 = D6 × Z2 Z6, D6 (2), Z22 (3), Z3, Z2 (7) Dihedral group, Dih6, product.
14 26 G141 D14 Z7, Z2 (7) Dihedral group, Dih7, Frobenius group
16[5] 31 G163 G4,4 = K4 ⋊ Z4 Z23, Z4 × Z2 (2), Z4 (4), Z22 (7), Z2 (7) Has the same number of elements of every order as the Pauli group. Nilpotent.
32 G164 Z4 ⋊ Z4 Z22 × Z2 (3), Z4 (6), Z22, 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 G166 Z8 ⋊ Z2 Z8 (2), Z22 × Z2, Z4 (2), Z22, 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 G167 D16 Z8, D8 (2), Z22 (4), Z4, Z2 (9) Dihedral group, Dih8. Nilpotent.
36 G168 QD16 Z8, Q8, D8, Z4 (3), Z22 (2), Z2 (5) The order 16 quasidihedral group. Nilpotent.
37 G169 Q16 Z8, Q8 (2), Z4 (5), Z2 Generalized quaternion group, Dicyclic group Dic4, binary dihedral group, <4,2,2>.[4] Nilpotent.
39 G1611 D8 × Z2 D8 (4), Z4 × Z2, Z23 (2), Z22 (13), Z4 (2), Z2 (11) Product. Nilpotent.
40 G1612 Q8 × Z2 Q8 (4), Z22 × Z2 (3), Z4 (6), Z22, Z2 (3) Hamiltonian group, product. Nilpotent.
41 G1613 (Z4 × Z2) ⋊ Z2 Q8, D8 (3), Z4 × Z2 (3), Z4 (4), Z22 (3), Z2 (7) The Pauli group generated by the Pauli matrices. Nilpotent.
18 44 G181 D18 Z9, D6 (3), Z3, Z2 (9) Dihedral group, Dih9, Frobenius group.
46 G183 Z3 ⋊ Z6 = D6 × Z3 = S3 × Z3 Z32, D6, Z6 (3), Z3 (4), Z2 (3) Product.
47 G184 (Z3 × Z3) ⋊ Z2 Z32, D6 (12), Z3 (4), Z2 (9) Frobenius group.
20 50 G201 Q20 Z10, Z5, Z4 (5), Z2 Dicyclic group Dic5, Binary dihedral group, <5,2,2>.[4]
52 G203 Z5 ⋊ Z4 D10, Z5, Z4 (5), Z2 (5) Frobenius group.
53 G204 D20 = D10 × Z2 Z10, D10 (2), Z5, Z22 (5), Z2 (11) Dihedral group, Dih10, product.
21 55 G211 Z7 ⋊ Z3 Z7, Z3 (7) Smallest non-abelian group of odd order. Frobenius group.
22 57 G221 D22 Z11, Z2 (11) Dihedral group Dih11, Frobenius group.
24 60 G241 Z3 ⋊ Z8 Z12, Z8 (3), Z6, Z4, Z3, Z2 Central extension of S3.
62 G243 SL(2,3) = Q8 ⋊ Z3 Q8, Z6 (4), Z4 (3), Z3 (4), Z2 Binary tetrahedral group, 2T = <3,3,2>.[4]
63 G244 Q24 = Z3 ⋊ Q8 Z12, Q12 (2), Q8 (3), Z6, Z4 (7), Z3, Z2 Dicyclic group Dic6, Binary dihedral, <6,2,2>.[4]
64 G245 D6 × Z4 = S3 × Z4 Z12, D12, Q12, Z4 × Z2 (3), Z6, D6 (2), Z4 (4), Z22 (3), Z3, Z2 (7) Product.
65 G246 D24 Z12, D12 (2), D8 (3), Z6, D6 (4), Z4, Z22 (6), Z3, Z2 (13) Dihedral group, Dih12.
66 G247 Q12 × Z2 = Z2 × (Z3 ⋊ Z4) Z6 × Z2, Q12 (2), Z4 × Z2 (3), Z6 (3), Z4 (6), Z22, Z3, Z2 (3) Product.
67 G248 (Z6 × Z2) ⋊ Z2 = Z3 ⋊ Dih4 Z6 × Z2, D12, Q12, D8 (3), Z6 (3), D6 (2), Z4 (3), Z22 (4), Z3, Z2 (9) Double cover of dihedral group.
69 G2410 D8 × Z3 Z12, Z6 × Z2 (2), D8, Z6 (5), Z4, Z22 (2), Z3, Z2 (5) Product. Nilpotent.
70 G2411 Q8 × Z3 Z12 (3), Q8, Z6, Z4 (3), Z3, Z2 Product. Nilpotent.
71 G2412 S4 A4, D8 (3), D6 (4), Z4 (3), Z22 (4), Z3 (4), Z2 (9)[6] Symmetric group. Has no normal Sylow subgroups. Chiral octahedral symmetry (O), Achiral tetrahedral symmetry (Td)
72 G2413 A4 × Z2 A4, Z23, Z6 (4), Z22 (7), Z3 (4), Z2 (7) Product. Pyritohedral symmetry (Th)
73 G2414 D12 × Z2 Z6 × Z2, D12 (6), Z23 (3), Z6 (3), D6 (4), Z22 (19), Z3, Z2 (15) Product.
26 77 G261 D26 Z13, Z2 (13) Dihedral group, Dih13, Frobenius group.
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.
28 84 G281 Z7 ⋊ Z4 Z14, Z7, Z4 (7), Z2 Dicyclic group Dic7, Binary dihedral group, <7,2,2>.[4]
86 G283 D28 = D14 × Z2 Z14, D14 (2), Z7, Z22 (7), Z2 (9) Dihedral group, Dih14, product.
30 89 G301 D6 × Z5 Z15, Z10 (3), D6, Z5, Z3, Z2 (3) Product.
90 G302 D10 × Z3 Z15, D10, Z6 (5), Z5, Z3, Z2 (5) Product.
91 G303 D30 Z15, D10 (3), D6 (5), Z5, Z3, Z2 (15) Dihedral group, Dih15, Frobenius group.

Classifying groups of small order

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

  • Order p: The only group is cyclic.
  • Order p2: There are just two groups, both abelian.
  • Order p3: 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 p2 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 p4: 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 = 211.[7]

Small Groups Library

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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:[8]

  • those of order at most 2000[9] 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[10]);
  • 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

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Notes

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  1. ^ a b Identifier when groups are numbered by order, o, then by index, i, from the small groups library, starting at 1.
  1. ^ a b 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. ^ a b c d e f g 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>: Rl=Sm=Tn=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

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  • 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 2n (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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