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[edit]

(sequence A000001 in OEIS)

Total number of nonisomorphic groups by order
+ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0 0 1 1 1 2 1 2 1 5 2 2 1 5 1 2 1
16 14 1 5 1 5 2 2 1 15 2 2 5 4 1 4 1
32 51 1 2 1 14 1 2 2 14 1 6 1 4 2 2 1
48 52 2 5 1 5 1 15 2 13 2 2 1 13 1 2 4
64 267 1 4 1 5 1 4 1 50 1 2 3 4 1 6 1
80 52 15 2 1 15 1 2 1 12 1 10 1 4 2 2 1
96 231 1 5 2 16 1 4 1 14 2 2 1 45 1 6 2
112 43 1 6 1 5 4 2 1 47 2 2 1 4 5 16 1
128 2328 2 4 1 10 1 2 5 15 1 4 1 11 1 2 1

For labeled groups, see OEISA034383.

Glossary[edit]

Each group is named by their Small Groups library index as Goi, where o is the order of the group, and i is the index of 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. GH 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 multiple isomorphic subgroups, their number is indicated in parentheses.

List of small abelian groups[edit]

The finite abelian groups are either cyclic groups, or direct products thereof; see abelian groups.

(sequence A000688 in OEIS)

Number of nonisomorphic abelian groups by order
+ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0 0 1 1 1 2 1 1 1 3 2 1 1 2 1 1 1
16 5 1 2 1 2 1 1 1 3 2 1 3 2 1 1 1
32 7 1 1 1 4 1 1 1 3 1 1 1 2 2 1 1
48 5 2 2 1 2 1 3 1 3 1 1 1 2 1 1 2
64 11 1 1 1 2 1 1 1 6 1 1 2 2 1 1 1
80 5 5 1 1 2 1 1 1 3 1 2 1 2 1 1 1
96 7 1 2 2 4 1 1 1 3 1 1 1 6 1 1 1
112 5 1 1 1 2 2 1 1 3 2 1 1 2 3 2 1
128 15 1 1 1 2 1 1 3 3 1 1 1 2 1 1 1

For labeled Abelian groups, see OEISA034382.

List of all abelian groups up to order 30
Order Goi Group Subgroups Cycle
graph
Properties
1 [1] G11 Z1[2] = S1 = A2 GroupDiagramMiniC1.svg Trivial group. Cyclic, alternating group, symmetric group. Elementary.
2 [3] G21 Z2[4] = S2 = Dih1 GroupDiagramMiniC2.svg simple, the smallest non-trivial group. Symmetric group. Cyclic. Elementary.
3 [5] G31 Z3[6] = A3 GroupDiagramMiniC3.svg simple. Alternating group. Cyclic. Elementary.
4[7] G41 Z4[8] = Dic1 Z2 GroupDiagramMiniC4.svg cyclic.
G42 Z22 = K4[9] = Dih2 Z2 (3) GroupDiagramMiniD4.svg Klein four-group, the smallest non-cyclic group. Elementary. Product.
5[10] G51 Z5[11] GroupDiagramMiniC5.svg Simple. Cyclic. Elementary.
6[12] G62 Z6[13] = Z3 × Z2[14] Z3, Z2 GroupDiagramMiniC6.svg Cyclic. Product.
7[15] G71 Z7[16] GroupDiagramMiniC7.svg Simple. Cyclic. Elementary.
8[17] G81 Z8[18] Z4, Z2 GroupDiagramMiniC8.svg Cyclic.
G82 Z4 × Z2[19] Z22, Z4 (2), Z2 (3) GroupDiagramMiniC2C4.svg Product.
G85 Z23[20] Z22 (7), Z2 (7) GroupDiagramMiniC2x3.svg The non-identity elements correspond to the points in the Fano plane, the Z2 × Z2 subgroups to the lines. Product. Elementary.
9[21] G91 Z9[22] Z3 GroupDiagramMiniC9.svg Cyclic.
G92 Z32[23] Z3 (4) GroupDiagramMiniC3x2.svg Elementary. Product.
10[24] G102 Z10[25] = Z5 × Z2 Z5, Z2 GroupDiagramMiniC10.svg Cyclic. Product.
11 G111 Z11[26] GroupDiagramMiniC11.svg Simple. Cyclic. Elementary.
12[27] G122 Z12[28] = Z4 × Z3 Z6, Z4, Z3, Z2 GroupDiagramMiniC12.svg Cyclic. Product.
G125 Z6 × Z2[29] = Z3 × K4 Z6 (3), Z3, Z2 (3), Z22 GroupDiagramMiniC2C6.svg Product.
13 G131 Z13[30] GroupDiagramMiniC13.svg Simple. Cyclic. Elementary.
14[31] G142 Z14[32] = Z7 × Z2 Z7, Z2 GroupDiagramMiniC14.svg Cyclic. Product.
15[33] G151 Z15[34] = Z5 × Z3 Z5, Z3 GroupDiagramMiniC15.svg Cyclic. Product.
16[35] G161 Z16[36] Z8, Z4, Z2 GroupDiagramMiniC16.svg Cyclic.
G162 Z42[37] Z2 (3), Z4 (6), Z22, Z4 × Z2 (3) GroupDiagramMiniC4x2.svg Product.
G165 Z8 × Z2[38] Z2 (3), Z4 (2), Z22, Z8 (2), Z4 × Z2 GroupDiagramC2C8.svg Product.
G1610 Z4 × K4[39] Z2 (7), Z4 (4), Z22 (7), Z23, Z4 × Z2 (6) GroupDiagramMiniC2x2C4.svg Product.
G1614 Z24[19] = K42 Z2 (15), Z22 (35), Z23 (15) GroupDiagramMiniC2x4.svg Product. Elementary.
17 G171 Z17[40] GroupDiagramMiniC17.svg Simple. Cyclic. Elementary.
18[41] G182 Z18[42] = Z9 × Z2 Z9, Z6, Z3, Z2 GroupDiagramMiniC18.svg Cyclic. Product.
G185 Z6 × Z3[43] = Z32 × Z2 Z6, Z3, Z2 GroupDiagramMiniC3C6.png Product.
19 G191 Z19[44] GroupDiagramMiniC19.svg Simple. Cyclic. Elementary.
20[45] G202 Z20[46] = Z5 × Z4 Z20, Z10, Z5, Z4, Z2 GroupDiagramMiniC20.svg Cyclic. Product.
G205 Z10 × Z2[47] = Z5 × Z22 Z5, Z2 GroupDiagramMiniC2C10.png Product.
21 G212 Z21[48] = Z7 × Z3 Z7, Z3 GroupDiagramMiniC21.svg Cyclic. Product.
22 G222 Z22[49] = Z11 × Z2 Z11, Z2 GroupDiagramMiniC22.svg Cyclic. Product.
23 G231 Z23[50] GroupDiagramMiniC23.svg Simple. Cyclic. Elementary.
24[51] G242 Z24[52] = Z8 × Z3 Z12, Z8, Z6, Z4, Z3, Z2 GroupDiagramMiniC24.svg Cyclic. Product.
G249 Z12 × Z2[53] = Z6 × Z4
= Z4 × Z3 × Z2
Z12, Z6, Z4, Z3, Z2 Product.
G2415 Z6 × Z22[39] Z6, Z3, Z2 Product.
25 G251 Z25 Z5 Cyclic.
G252 Z52 Z5 Product. Elementary.
26 G261 Z26 = Z13 × Z2 Z13, Z2 Cyclic. Product.
27[54] G271 Z27 Z9, Z3 Cyclic.
G272 Z9 × Z3 Z9, Z3 Product.
G27 Z33 Z3 Product. Elementary.
28 G282 Z28 = Z7 × Z4 Z14, Z7, Z4, Z2 Cyclic. Product.
G284 Z14 × Z2 = Z7 × Z22 Z14, Z7, Z4, Z2 Product.
29 G291 Z29 Simple. Cyclic. Elementary.
30[55] G304 Z30 = Z15 × Z2 = Z10 × Z3
= Z6 × Z5 = Z5 × Z3 × Z2
Z15, Z10, Z6, Z5, Z3, Z2 Cyclic. Product.

List of small non-abelian groups[edit]

(sequence A060689 in OEIS)

Number of nonisomorphic nonabelian groups by order
+ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0 0 0 0 0 0 0 1 0 2 0 1 0 3 0 1 0
16 9 0 3 0 3 1 1 0 12 0 1 2 2 0 3 0
32 44 0 1 0 10 0 1 1 11 0 5 0 2 0 1 0
48 47 0 3 0 3 0 12 1 10 1 1 0 11 0 1 2
64 256 0 3 0 3 0 3 0 44 0 1 1 2 0 5 0
80 47 10 1 0 13 0 1 0 9 0 8 0 2 1 1 0
96 224 0 3 0 12 0 3 0 11 1 1 0 39 0 5 1
112 38 0 5 0 3 2 1 0 44 0 1 0 2 2 14 0
128 2313 1 3 0 8 0 1 2 12 0 3 0 9 0 1 0

Order of non-abelian groups 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, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 66, 68, 70, 72, 74, 75, 76, 78, 80, 81, 82, 84, 86, 88, 90, 92, 93, 94, 96, 98, 100, 102, 104, 105, 106, 108, 110, 111, 112, 114, 116, 117, 118, 120, 122, 124, 125, 126, 128, 129, 130, 132, 134, 135, 136, 138, 140, 142, ... (sequence A060652 in OEIS)
List of all nonabelian groups up to order 30
Order Goi Group Subgroups Cycle
graph
Properties
6[12] G61 Dih3 = S3 Z3, Z2 (3) GroupDiagramMiniD6.svg Dihedral group, the smallest non-abelian group, symmetric group, Frobenius group
8[17] G83 Dih4 Z4, Z22 (2), Z2 (5) GroupDiagramMiniD8.svg Dihedral group. Extraspecial group. Nilpotent.
G84 Q8 = Dic2 = <2,2,2> Z4 (3), Z2 GroupDiagramMiniQ8.svg 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 Binary dihedral group. Nilpotent.
10[24] G101 Dih5 Z5, Z2 (5) GroupDiagramMiniD10.svg Dihedral group, Frobenius group
12[27] G121 Q12 = Dic3 = <3,2,2>
= Z3 ⋊ Z4
Z2, Z3, Z4 (3), Z6 GroupDiagramMiniX12.svg Binary dihedral group
G123 A4 Z22, Z3 (4), Z2 (3) GroupDiagramMiniA4.svg Alternating group. No subgroup of order 6, although 6 divides its order. Frobenius group
G124 Dih6 = Dih3 × Z2 Z6, Dih3 (2), Z22 (3), Z3, Z2 (7) GroupDiagramMiniD12.svg Dihedral group, product
14[31] G141 Dih7 Z7, Z2 (7) GroupDiagramMiniD14.svg Dihedral group, Frobenius group
16[35][56] G163 G4,4 = K4 ⋊ Z4
(Z4×Z2) ⋊ Z2
GroupDiagramMiniG44.svg Has the same number of elements of every order as the Pauli group. Nilpotent.
G164 Z4 ⋊ Z4 GroupDiagramMinix3.svg The squares of elements do not form a subgroup. Has the same number of elements of every order as Q8 × Z2. Nilpotent.
G166 Z8 ⋊ Z2 GroupDiagramMOD16.svg Sometimes called the modular group of order 16, though this is misleading as abelian groups and Q8 × Z2 are also modular. Nilpotent.
G167 Dih8 Z8, Dih4 (2), Z22 (4), Z4, Z2 (9) GroupDiagramMiniD16.svg Dihedral group. Nilpotent.
G168 QD16 GroupDiagramMiniQH16.svg The order 16 quasidihedral group. Nilpotent.
G169 Q16 = Dic4 = <4,2,2> GroupDiagramMiniQ16.svg generalized quaternion group, binary dihedral group. Nilpotent.
G1611 Dih4 × Z2 Dih4 (2), Z4 × Z2, Z23 (2), Z22 (11), Z4 (2), Z2 (11) GroupDiagramMiniC2D8.svg Product. Nilpotent.
G1612 Q8 × Z2 GroupDiagramMiniC2Q8.svg Hamiltonian, product. Nilpotent.
G1613 (Z4 × Z2) ⋊ Z2 GroupDiagramMiniC2x2C4.svg The Pauli group generated by the Pauli matrices. Nilpotent.
18[41] G181 Dih9 GroupDiagramMiniD18.png Dihedral group, Frobenius group
G183 S3×Z3 GroupDiagramMiniC3D6.png Product
G184 (Z3 × Z3)⋊ Z2 GroupDiagramMiniG18-4.png Frobenius group
20[45] G201 Q20 = Dic5 = <5,2,2> GroupDiagramMiniQ20.png Binary dihedral group
G203 Z5 ⋊ Z4 GroupDiagramMiniC5semiprodC4.png Frobenius group
G204 Dih10 = Dih5 × Z2 GroupDiagramMiniD20.png Dihedral group, product
21 G211 Z7 ⋊ Z3 Smallest non-abelian group of odd order. Frobenius group
22 G221 Dih11 Dihedral group, Frobenius group
24[51] G241 Z3 ⋊ Z8 Central extension of S3
G243 SL(2,3) = 2T = Q8 ⋊ Z3 SL(2,3); Cycle graph.svg Binary tetrahedral group
G244 Q24 = Dic6 = <6,2,2> = Z3 ⋊ Q8 GroupDiagramMiniQ24.png Binary dihedral
G245 Z4 × S3 Product
G246 Dih12 Dihedral group
G247 Dic3 × Z2 = Z2 × (Z3 × Z4) Product
G248 (Z6 × Z2)⋊ Z2 = Z3 ⋊ Dih4 Double cover of dihedral group
G2410 Dih4 × Z3 Product. Nilpotent.
G2411 Q8 × Z3 Product. Nilpotent.
G2412 S4 Symmetric group 4; cycle graph.svg Symmetric group. Has no normal Sylow subgroups.
G2413 A4 × Z2 GroupDiagramMiniA4xC2.png Product
G2414 D12× Z2 Product
26 G261 Dih13 Dihedral group, Frobenius group
27[54] G273 Z32 ⋊ Z3 All non-trivial elements have order 3. Extraspecial group. Nilpotent.
G274 Z9 ⋊ Z3 Extraspecial group. Nilpotent.
28 G281 Z7 ⋊ Z4 Binary dihedral group
G283 Dih14 Dihedral group, product
30[55] G301 Z5 × S3 Product
G303 Dih15 Dihedral group, Frobenius group
G304 Z3 × Dih5 Product

More information[edit]

Groups of order 32, 36, 40, 48, 54, 56, 60, 64, 72.

Classifying groups of small order[edit]

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 × Z/2Z
  • Order 60: The alternating group A5.

Small groups library[edit]

The group theoretical computer algebra system GAP contains 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:[57]

  • 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 an additional 49487365422 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 qn × p 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.

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

See also[edit]

Notes[edit]

  1. ^ Groups of order 1
  2. ^ Z1
  3. ^ Groups of order 2
  4. ^ Z2
  5. ^ Groups of order 3
  6. ^ Z3
  7. ^ Groups of order 4
  8. ^ Z4
  9. ^ Klein group
  10. ^ Groups of order 5
  11. ^ Z5
  12. ^ a b Groups of order 6
  13. ^ Z6
  14. ^ See a worked example showing the isomorphism Z6 = Z3 × Z2.
  15. ^ Groups of order 7
  16. ^ Z7
  17. ^ a b Groups of order 8
  18. ^ Z8
  19. ^ a b Z4×Z2
  20. ^ Elementary abelian group:E8
  21. ^ Groups of order 9
  22. ^ Z9
  23. ^ Z3×Z3
  24. ^ a b Groups of order 10
  25. ^ Z10
  26. ^ Z11
  27. ^ a b Groups of order 12
  28. ^ Z12
  29. ^ Z6×Z2
  30. ^ Z13
  31. ^ a b Groups of order 14
  32. ^ Z14
  33. ^ Groups of order 15
  34. ^ Z15
  35. ^ a b Groups of order 16
  36. ^ Z16
  37. ^ Z4×Z4
  38. ^ Z8×Z2
  39. ^ a b Z4×Z2×Z2
  40. ^ Z17
  41. ^ a b Groups of order 18
  42. ^ Z18
  43. ^ Z6×Z3
  44. ^ Z19
  45. ^ a b Groups of order 20
  46. ^ Z20
  47. ^ Z10×Z2
  48. ^ Z21
  49. ^ Z22
  50. ^ Z23
  51. ^ a b Groups of order 24
  52. ^ Z24
  53. ^ Z12×Z2
  54. ^ a b Groups of order 27
  55. ^ a b Groups of order 30
  56. ^ Wild, Marcel. "The Groups of Order Sixteen Made Easy", American Mathematical Monthly, Jan 2005
  57. ^ Hans Ulrich Besche The Small Groups library

References[edit]

  • Coxeter, H. S. M. and 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)". Macmillan. MR 168631. 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[edit]