List of small groups

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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.

Counts[edit]

Total number of nonisomorphic groups by order[1]
Add 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
0 1 1 1 2 1 2 1 5 2 2 1 5 1 2 1 14 1 5 1 5 2 2 1 15
24 2 2 5 4 1 4 1 51 1 2 1 14 1 2 2 14 1 6 1 4 2 2 1 52
48 2 5 1 5 1 15 2 13 2 2 1 13 1 2 4 267 1 4 1 5 1 4 1 50
72 1 2 3 4 1 6 1 52 15 2 1 15 1 2 1 12 1 10 1 4 2

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; the particular action of H on G is not shown 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 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.

Number of nonisomorphic abelian groups by order[2]
Add 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
0 1 1 1 2 1 1 1 3 2 1 1 2 1 1 1 5 1 2 1 2 1 1 1 3
24 2 1 3 2 1 1 1 7 1 1 1 4 1 1 1 3 1 1 1 2 2 1 1 5
48 2 2 1 2 1 3 1 3 1 1 1 2 1 1 2 11 1 1 1 2 1 1 1 6
72 1 1 2 2 1 1 1 5 5 1 1 2 1 1 1 3 1 2 1 2 1 1 1 7
List of all abelian groups up to order 24
Order Goi Group Subgroups Cycle
graph
Properties
1 [3] G11 Z1[4] = S1 = A2 GroupDiagramMiniC1.svg trivial group, various properties hold trivially
2 [5] G21 Z2[6] = S2 = Dih1 GroupDiagramMiniC2.svg simple, the smallest non-trivial group
3 [7] G31 Z3[8] = A3 GroupDiagramMiniC3.svg simple
4[9] G41 Z4[10] Z2 GroupDiagramMiniC4.svg  
G42 Z 2
2
 
= K4[11] = Dih2
Z2 (3) GroupDiagramMiniD4.svg Klein four-group, K4, the smallest non-cyclic group
5[12] G51 Z5[13] GroupDiagramMiniC5.svg simple
6[14] G62 Z6[15] = Z3 × Z2[16] Z3, Z2 GroupDiagramMiniC6.svg  
7[17] G71 Z7[18] GroupDiagramMiniC7.svg simple
8[19] G81 Z8[20] Z4, Z2 GroupDiagramMiniC8.svg  
G82 Z4 × Z2[21] Z 2
2
 
, Z4 (2), Z2 (3)
GroupDiagramMiniC2C4.svg  
G85 Z 3
2
 
[22]
Z 2
2
 
(7), Z2 (7)
GroupDiagramMiniC2x3.svg the non-identity elements correspond to the points in the Fano plane, the Z2 × Z2 subgroups to the lines
9[23] G91 Z9[24] Z3 GroupDiagramMiniC9.svg  
G92 Z 2
3
 
[25]
Z3 (4) GroupDiagramMiniC3x2.svg  
10[26] G102 Z10[27] = Z5 × Z2 Z5, Z2 GroupDiagramMiniC10.svg  
11 G111 Z11[28] GroupDiagramMiniC11.svg simple
12[29] G122 Z12[30] = Z4 × Z3 Z6, Z4, Z3, Z2 GroupDiagramMiniC12.svg  
G125 Z6 × Z2[31] = Z3 × K4 Z6 (3), Z3, Z2 (3), Z 2
2
 
GroupDiagramMiniC2C6.svg  
13 G131 Z13[32] GroupDiagramMiniC13.svg simple
14[33] G142 Z14[34] = Z7 × Z2 Z7, Z2 GroupDiagramMiniC14.svg  
15[35] G151 Z15[36] = Z5 × Z3 Z5, Z3 GroupDiagramMiniC15.svg multiplication of nimbers 1,...,15
16[37] G161 Z16[38] Z8, Z4, Z2 GroupDiagramMiniC16.svg  
G162 Z 2
4
 
[39]
Z2 (3), Z4 (6), Z 2
2
 
, Z4 × Z2 (3)
GroupDiagramMiniC4x2.svg  
G165 Z8 × Z2[40] Z2 (3), Z4 (2), Z 2
2
 
, Z8 (2), Z4 × Z2
GroupDiagramC2C8.svg  
G1610 Z4 × K4[41] Z2 (7), Z4 (4), Z 2
2
 
(7), Z 3
2
 
, Z4 × Z2 (6)
GroupDiagramMiniC2x2C4.svg  
G1614 Z 4
2
 
[42] = K42
Z2 (15), Z 2
2
 
(35), Z 3
2
 
(15)
GroupDiagramMiniC2x4.svg addition of nimbers 0,...,15
17 G171 Z17[43] GroupDiagramMiniC17.svg simple
18[44] G182 Z18[45] = Z9 × Z2 Z9, Z6, Z3, Z2 GroupDiagramMiniC18.svg
G185 Z6 × Z3[46] = Z32 × Z2 Z6, Z3, Z2 GroupDiagramMiniC3C6.png
19 G191 Z19[47] GroupDiagramMiniC19.svg simple
20[48] G202 Z20[49] = Z5 × Z4 Z20, Z10, Z5, Z4, Z2 GroupDiagramMiniC20.svg
G205 Z10 × Z2[50] = Z5 × Z22 Z5, Z2 GroupDiagramMiniC2C10.png
21 G212 Z21[51] = Z7 × Z3 Z7, Z3 GroupDiagramMiniC21.svg
22 G222 Z22[52] = Z11 × Z2 Z11, Z2 GroupDiagramMiniC22.svg
23 G231 Z23[53] GroupDiagramMiniC23.svg simple
24[54] G242 Z24[55] = Z8 × Z3 Z12, Z8, Z6, Z4, Z3, Z2 GroupDiagramMiniC23.svg
G249 Z12 × Z2[56] = Z6 × Z4 Z12, Z6, Z4, Z3, Z2
G2415 Z6 × Z22[57] Z6, Z3, Z2

List of small non-abelian groups[edit]

Number of nonisomorphic nonabelian groups by order[58]
Add 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
0 0 0 0 0 0 1 0 2 0 1 0 3 0 1 0 9 0 3 0 3 1 1 0 12
24 0 1 2 2 0 3 0 44 0 1 0 10 0 1 1 11 0 5 0 2 0 1 0 47
48 0 3 0 3 0 12 1 10 1 1 0 11 0 1 2 256 0 3 0 3 0 3 0 44
72 0 1 1 2 0 5 0 47 10 1 0 13 0 1 0 9 0 8 0 2 1
List of all nonabelian groups up to order 16
Order Goi Group Subgroups Cycle
graph
Properties
6[14] G61 Dih3 = S3 Z3, Z2 (3) GroupDiagramMiniD6.svg Dihedral group, the smallest non-abelian group
8[19] G83 Dih4 Z4, Z22 (2), Z2 (5) GroupDiagramMiniD8.svg Dihedral group
G84 Q8 = Dic2 Z4 (3), Z2 GroupDiagramMiniQ8.svg Quaternion group, the smallest Hamiltonian group; smallest group demonstrating that all subgroups may be 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
10[26] G101 Dih5 Z5, Z2 (5) GroupDiagramMiniD10.svg Dihedral group
12[29] G121 Dic3 = Q12 = Z3 ⋊ Z4 Z2, Z3, Z4 (3), Z6 GroupDiagramMiniX12.svg
G123 A4 Z22, Z3 (4), Z2 (3) GroupDiagramMiniA4.svg 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.)
G124 Dih6 = Dih3 × Z2 Z6, Dih3 (2), Z22 (3), Z3, Z2 (7) GroupDiagramMiniD12.svg Dihedral group
14[33] G141 Dih7 Z7, Z2 (7) GroupDiagramMiniD14.svg Dihedral group
16[37][59] G163 G4,4 = K4 ⋊ Z4
(Z4×Z2) ⋊ Z2
GroupDiagramMiniG44.svg
G164 Z4 ⋊ Z4 GroupDiagramMinix3.svg
G166 Z8 ⋊ Z2 GroupDiagramMOD16.svg The order 16 modular group
G167 Dih8 Z8, Dih4 (2), Z22 (4), Z4, Z2 (9) GroupDiagramMiniD16.svg Dihedral group
G168 QD16 GroupDiagramMiniQH16.svg The order 16 quasidihedral group
G169 Dic4 = Q16 GroupDiagramMiniQ16.svg generalized quaternion group
G1611 Dih4 × Z2 Dih4 (2), Z4 × Z2, Z23 (2), Z22 (11), Z4 (2), Z2 (11) GroupDiagramMiniC2D8.svg
G1612 Q8 × Z2 GroupDiagramMiniC2Q8.svg Hamiltonian
G1613 (Z4 × Z2) ⋊ Z2 GroupDiagramMiniC2x2C4.svg The group generated by the Pauli matrices
18[44] G181 Dih9 GroupDiagramMiniD18.png Dihedral group
G183 S3×Z3 GroupDiagramMiniC3D6.png
G184 (Z3 × Z3)⋊ Z2 GroupDiagramMiniG18-4.png
20[48] G201 Dic5 = Q20 GroupDiagramMiniQ20.png
G203 Z5 ⋊ Z4 GroupDiagramMiniC5semiprodC4.png
G204 Dih10 = Dih5 × Z2 GroupDiagramMiniD20.png Dihedral group
21 G211 Z7 ⋊ Z3
22 G221 Dih11 Dihedral group
24[54] G241 Z3 ⋊ Z8
G243 SL(2,3) SL(2,3); Cycle graph.svg
G244 Dic6 = Q24 = Z3 ⋊ Q8 GroupDiagramMiniQ24.png
G245 Z4 ⋊ S3
G246 Dih12 Dihedral group
G247 Dic3 × Z2 = Z2 × (Z3 × Z4)
G248 (Z6 × Z2)⋊ Z2 = Z3 ⋊ Dih4
G249 Z12 × Z2
G2410 Z8 × Z3
G2411 Q8 × Z3
G2412 S4 Symmetric group 4; cycle graph.svg Symmetric group
G2413 A4 × Z2 GroupDiagramMiniA4xC2.png

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:[60]

  • 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. ^ OEISA000001
  2. ^ OEISA000688
  3. ^ Groups of order 1
  4. ^ Z1
  5. ^ Groups of order 2
  6. ^ Z2
  7. ^ Groups of order 3
  8. ^ Z3
  9. ^ Groups of order 4
  10. ^ Z4
  11. ^ Klein group
  12. ^ Groups of order 5
  13. ^ Z5
  14. ^ a b Groups of order 6
  15. ^ Z6
  16. ^ See a worked example showing the isomorphism Z6 = Z3 × Z2.
  17. ^ Groups of order 7
  18. ^ Z7
  19. ^ a b Groups of order 8
  20. ^ Z8
  21. ^ Z4×Z2
  22. ^ Elementary abelian group:E8
  23. ^ Groups of order 9
  24. ^ Z9
  25. ^ Z3×Z3
  26. ^ a b Groups of order 10
  27. ^ Z10
  28. ^ Z11
  29. ^ a b Groups of order 12
  30. ^ Z12
  31. ^ Z6×Z2
  32. ^ Z13
  33. ^ a b Groups of order 14
  34. ^ Z14
  35. ^ Groups of order 15
  36. ^ Z15
  37. ^ a b Groups of order 16
  38. ^ Z16
  39. ^ Z4×Z4
  40. ^ Z8×Z2
  41. ^ Z4×Z2×Z2
  42. ^ Z4×Z2
  43. ^ Z17
  44. ^ a b Groups of order 18
  45. ^ Z18
  46. ^ Z6×Z3
  47. ^ Z19
  48. ^ a b Groups of order 20
  49. ^ Z20
  50. ^ Z10×Z2
  51. ^ Z21
  52. ^ Z22
  53. ^ Z23
  54. ^ a b Groups of order 24
  55. ^ Z24
  56. ^ Z12×Z2
  57. ^ Z4×Z2×Z2
  58. ^ OEISA046056
  59. ^ Wild, Marcel. "The Groups of Order Sixteen Made Easy", American Mathematical Monthly, Jan 2005
  60. ^ Hans Ulrich Besche The Small Groups library

References[edit]