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]

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

Number of nonisomorphic abelian groups by order[2]
Add 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
0 0 1 1 1 2 1 1 1 3 2 1 1 2 1 1 1 5 1 2 1 2 1 1 1
24 3 2 1 3 2 1 1 1 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 11 1 1 1 2 1 1 1
72 6 1 1 2 2 1 1 1 5 5 1 1 2 1 1 1 3 1 2 1 2 1 1 1
List of all abelian groups up to order 30
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 = Dic1[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 GroupDiagramMiniC24.svg
G249 Z12 × Z2[56] = Z6 × Z4
= Z4 × Z3 × Z2
Z12, Z6, Z4, Z3, Z2
G2415 Z6 × Z22[57] Z6, Z3, Z2
25 G251 Z25 Z5
G252 Z52 Z5
26 G261 Z26 = Z13 × Z2 Z13, Z2
27[58] G271 Z27 Z9, Z3
G272 Z9×Z3 Z9, Z3
G27 Z33 Z3
28 G282 Z28 = Z7 × Z4 Z14, Z7, Z4, Z2
G284 Z14 × Z2 = Z7 × Z22 Z14, Z7, Z4, Z2
29 G291 Z29 simple
30[59] G304 Z30 = Z15 × Z2 = Z10 × Z3
= Z6 × Z5 = Z5 × Z3 × Z2
Z15, Z10, Z6, Z5, Z3, Z2

List of small non-abelian groups[edit]

Number of nonisomorphic nonabelian groups by order[60]
Add 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
0 0 0 0 0 0 0 1 0 2 0 1 0 3 0 1 0 9 0 3 0 3 1 1 0
24 12 0 1 2 2 0 3 0 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 2 256 0 3 0 3 0 3 0
72 44 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 30
Order Goi Group Subgroups Cycle
graph
Properties
6[14] G61 Dih3 = S3 Z3, Z2 (3) GroupDiagramMiniD6.svg Dihedral group, the smallest non-abelian group, symmetric group, Frobenius group
8[19] G83 Dih4 Z4, Z22 (2), Z2 (5) GroupDiagramMiniD8.svg Dihedral group. Extraspecial group
G84 Q8 = Dic2 = <2,2,2> 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. Extraspecial group
10[26] G101 Dih5 Z5, Z2 (5) GroupDiagramMiniD10.svg Dihedral group, Frobenius group
12[29] 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 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.) Frobenius group
G124 Dih6 = Dih3 × Z2 Z6, Dih3 (2), Z22 (3), Z3, Z2 (7) GroupDiagramMiniD12.svg Dihedral group, product
14[33] G141 Dih7 Z7, Z2 (7) GroupDiagramMiniD14.svg Dihedral group, Frobenius group
16[37][61] G163 G4,4 = K4 ⋊ Z4
(Z4×Z2) ⋊ Z2
GroupDiagramMiniG44.svg Has the same number of elements of every order as the Pauli group.
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
G166 Z8 ⋊ Z2 GroupDiagramMOD16.svg The non-abelian 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 Q16 = Dic4 = <4,2,2> GroupDiagramMiniQ16.svg generalized quaternion group, binary dihedral group
G1611 Dih4 × Z2 Dih4 (2), Z4 × Z2, Z23 (2), Z22 (11), Z4 (2), Z2 (11) GroupDiagramMiniC2D8.svg Product
G1612 Q8 × Z2 GroupDiagramMiniC2Q8.svg Hamiltonian, product
G1613 (Z4 × Z2) ⋊ Z2 GroupDiagramMiniC2x2C4.svg The Pauli group generated by the Pauli matrices
18[44] G181 Dih9 GroupDiagramMiniD18.png Dihedral group, Frobenius group
G183 S3×Z3 GroupDiagramMiniC3D6.png Product
G184 (Z3 × Z3)⋊ Z2 GroupDiagramMiniG18-4.png Frobenius group
20[48] 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[54] 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
G2411 Q8 × Z3 Product
G2412 S4 Symmetric group 4; cycle graph.svg Symmetric group
G2413 A4 × Z2 GroupDiagramMiniA4xC2.png Product
G2414 D12× Z2 Product
26 G261 Dih13 Dihedral group, Frobenius group
27[58] G273 Z32 ⋊ Z3 All non-trivial elements have order 3. Extraspecial group
G274 Z9 ⋊ Z3 Extraspecial group
28 G281 Z7 ⋊ Z4 Binary dihedral group
G283 Dih14 Dihedral group, product
30[59] G301 Z5 × S3 Product
G303 Dih15 Dihedral group, Frobenius group
G304 Z3 × Dih5 Product

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:[62]

  • 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. ^ a b Groups of order 27
  59. ^ a b Groups of order 30
  60. ^ OEISA060689
  61. ^ Wild, Marcel. "The Groups of Order Sixteen Made Easy", American Mathematical Monthly, Jan 2005
  62. ^ Hans Ulrich Besche The Small Groups library

References[edit]