List of small groups

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

Counts

Glossary

Each group is named by their Small Groups library index as G_oⁱ, where o is the order of the group, and i is the index of the group within that order.

Common group names:

• Z_n: the cyclic group of order n (the notation C_n is also used; it is isomorphic to the additive group of Z/nZ).
  • K₄: the Klein four-group of order 4, same as Z₂ × Z₂ or Dih₂.
• Dih_n: the dihedral group of order 2n (often the notation D_n or D_2n is used )
• S_n: the symmetric group of degree n, containing the n! permutations of n elements.
• A_n: the alternating group of degree n, containing the n!/2 even permutations of n elements.
• Dic_n or Q_4n: the dicyclic group of order 4n.
  • Q₈: the quaternion group of order 8, also Dic₂.

The notations Z_n and Dih_n have the advantage that point groups in three dimensions C_n and D_n 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 Z_n, 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

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

List of all abelian groups up to order 30

Order	Goi	Group	Subgroups	Cycle graph	Properties
1 [3]	G11	Z1[4] = S1 = A2	–		Trivial group. Cyclic, alternating group, symmetric group. Elementary.
2 [5]	G21	Z2[6] = S2 = Dih1	–		simple, the smallest non-trivial group. Symmetric group. Cyclic. Elementary.
3 [7]	G31	Z3[8] = A3	–		simple. Alternating group. Cyclic. Elementary.
4[9]	G41	Z4[10] = Dic1	Z2		cyclic.
	G42	Z22 = K4[11] = Dih2	Z2 (3)		Klein four-group, the smallest non-cyclic group. Elementary. Product.
5[12]	G51	Z5[13]	–		Simple. Cyclic. Elementary.
6[14]	G62	Z6[15] = Z3 × Z2[16]	Z3, Z2		Cyclic. Product.
7[17]	G71	Z7[18]	–		Simple. Cyclic. Elementary.
8[19]	G81	Z8[20]	Z4, Z2		Cyclic.
	G82	Z4 × Z2[21]	Z22, Z4 (2), Z2 (3)		Product.
	G85	Z23[22]	Z22 (7), Z2 (7)		The non-identity elements correspond to the points in the Fano plane, the Z2 × Z2 subgroups to the lines. Product. Elementary.
9[23]	G91	Z9[24]	Z3		Cyclic.
	G92	Z32[25]	Z3 (4)		Elementary. Product.
10[26]	G102	Z10[27] = Z5 × Z2	Z5, Z2		Cyclic. Product.
11	G111	Z11[28]	–		Simple. Cyclic. Elementary.
12[29]	G122	Z12[30] = Z4 × Z3	Z6, Z4, Z3, Z2		Cyclic. Product.
	G125	Z6 × Z2[31] = Z3 × K4	Z6 (3), Z3, Z2 (3), Z22		Product.
13	G131	Z13[32]	–		Simple. Cyclic. Elementary.
14[33]	G142	Z14[34] = Z7 × Z2	Z7, Z2		Cyclic. Product.
15[35]	G151	Z15[36] = Z5 × Z3	Z5, Z3		Cyclic. Product.
16[37]	G161	Z16[38]	Z8, Z4, Z2		Cyclic.
	G162	Z42[39]	Z2 (3), Z4 (6), Z22, Z4 × Z2 (3)		Product.
	G165	Z8 × Z2[40]	Z2 (3), Z4 (2), Z22, Z8 (2), Z4 × Z2		Product.
	G1610	Z4 × K4[41]	Z2 (7), Z4 (4), Z22 (7), Z23, Z4 × Z2 (6)		Product.
	G1614	Z24[21] = K42	Z2 (15), Z22 (35), Z23 (15)		Product. Elementary.
17	G171	Z17[42]	–		Simple. Cyclic. Elementary.
18[43]	G182	Z18[44] = Z9 × Z2	Z9, Z6, Z3, Z2		Cyclic. Product.
	G185	Z6 × Z3[45] = Z32 × Z2	Z6, Z3, Z2		Product.
19	G191	Z19[46]	–		Simple. Cyclic. Elementary.
20[47]	G202	Z20[48] = Z5 × Z4	Z20, Z10, Z5, Z4, Z2		Cyclic. Product.
	G205	Z10 × Z2[49] = Z5 × Z22	Z5, Z2		Product.
21	G212	Z21[50] = Z7 × Z3	Z7, Z3		Cyclic. Product.
22	G222	Z22[51] = Z11 × Z2	Z11, Z2		Cyclic. Product.
23	G231	Z23[52]	–		Simple. Cyclic. Elementary.
24[53]	G242	Z24[54] = Z8 × Z3	Z12, Z8, Z6, Z4, Z3, Z2		Cyclic. Product.
	G249	Z12 × Z2[55] = Z6 × Z4 = Z4 × Z3 × Z2	Z12, Z6, Z4, Z3, Z2		Product.
	G2415	Z6 × Z22[41]	Z6, Z3, Z2		Product.
25	G251	Z25	Z5		Cyclic.
	G252	Z52	Z5		Product. Elementary.
26	G261	Z26 = Z13 × Z2	Z13, Z2		Cyclic. Product.
27[56]	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[57]	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

List of all nonabelian groups up to order 30

Order	Goi	Group	Subgroups	Cycle graph	Properties
6[14]	G61	Dih3 = S3	Z3, Z2 (3)		Dihedral group, the smallest non-abelian group, symmetric group, Frobenius group
8[19]	G83	Dih4	Z4, Z22 (2), Z2 (5)		Dihedral group. Extraspecial group. Nilpotent.
	G84	Q8 = Dic2 = <2,2,2>	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 Binary dihedral group. Nilpotent.
10[26]	G101	Dih5	Z5, Z2 (5)		Dihedral group, Frobenius group
12[29]	G121	Q12 = Dic3 = <3,2,2> = Z3 ⋊ Z4	Z2, Z3, Z4 (3), Z6		Binary dihedral group
	G123	A4	Z22, Z3 (4), Z2 (3)		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)		Dihedral group, product
14[33]	G141	Dih7	Z7, Z2 (7)		Dihedral group, Frobenius group
16[37][59]	G163	G4,4 = K4 ⋊ Z4 (Z4×Z2) ⋊ Z2			Has the same number of elements of every order as the Pauli group. Nilpotent.
	G164	Z4 ⋊ Z4			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			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)		Dihedral group. Nilpotent.
	G168	QD16			The order 16 quasidihedral group. Nilpotent.
	G169	Q16 = Dic4 = <4,2,2>			generalized quaternion group, binary dihedral group. Nilpotent.
	G1611	Dih4 × Z2	Dih4 (2), Z4 × Z2, Z23 (2), Z22 (11), Z4 (2), Z2 (11)		Product. Nilpotent.
	G1612	Q8 × Z2			Hamiltonian, product. Nilpotent.
	G1613	(Z4 × Z2) ⋊ Z2			The Pauli group generated by the Pauli matrices. Nilpotent.
18[43]	G181	Dih9			Dihedral group, Frobenius group
	G183	S3×Z3			Product
	G184	(Z3 × Z3)⋊ Z2			Frobenius group
20[47]	G201	Q20 = Dic5 = <5,2,2>			Binary dihedral group
	G203	Z5 ⋊ Z4			Frobenius group
	G204	Dih10 = Dih5 × Z2			Dihedral group, product
21	G211	Z7 ⋊ Z3			Smallest non-abelian group of odd order. Frobenius group
22	G221	Dih11			Dihedral group, Frobenius group
24[53]	G241	Z3 ⋊ Z8			Central extension of S3
	G243	SL(2,3) = 2T = Q8 ⋊ Z3			Binary tetrahedral group
	G244	Q24 = Dic6 = <6,2,2> = Z3 ⋊ Q8			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. Has no normal Sylow subgroups.
	G2413	A4 × Z2			Product
	G2414	D12× Z2			Product
26	G261	Dih13			Dihedral group, Frobenius group
27[56]	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[57]	G301	Z5 × S3			Product
	G303	Dih15			Dihedral group, Frobenius group
	G304	Z3 × Dih5			Product

Classifying groups of small order

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

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

• Classification of finite simple groups
• Composition series
• List of finite simple groups
• Number of groups of a given order
• Small Latin squares and quasigroups

Notes

1. ↑  A000001
2. ↑  A000688
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. ↑ 14.0 14.1 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. ↑ 19.0 19.1 Groups of order 8
20. ↑ Z8
21. ↑ 21.0 21.1 Z4×Z2
22. ↑ Elementary abelian group:E8
23. ↑ Groups of order 9
24. ↑ Z9
25. ↑ Z3×Z3
26. ↑ 26.0 26.1 Groups of order 10
27. ↑ Z10
28. ↑ Z11
29. ↑ 29.0 29.1 Groups of order 12
30. ↑ Z12
31. ↑ Z6×Z2
32. ↑ Z13
33. ↑ 33.0 33.1 Groups of order 14
34. ↑ Z14
35. ↑ Groups of order 15
36. ↑ Z15
37. ↑ 37.0 37.1 Groups of order 16
38. ↑ Z16
39. ↑ Z4×Z4
40. ↑ Z8×Z2
41. ↑ 41.0 41.1 Z4×Z2×Z2
42. ↑ Z17
43. ↑ 43.0 43.1 Groups of order 18
44. ↑ Z18
45. ↑ Z6×Z3
46. ↑ Z19
47. ↑ 47.0 47.1 Groups of order 20
48. ↑ Z20
49. ↑ Z10×Z2
50. ↑ Z21
51. ↑ Z22
52. ↑ Z23
53. ↑ 53.0 53.1 Groups of order 24
54. ↑ Z24
55. ↑ Z12×Z2
56. ↑ 56.0 56.1 Groups of order 27
57. ↑ 57.0 57.1 Groups of order 30
58. ↑  A060689
59. ↑ Wild, Marcel. "The Groups of Order Sixteen Made Easy", American Mathematical Monthly, Jan 2005
60. ↑ Hans Ulrich Besche The Small Groups library

References

• 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

• Particular groups in the Group Properties Wiki
• Besche, H. U.; Eick, B.; O'Brien, E. "small group library". 
• R. J. Mathar (2014). "Plots of cycle graphs of the finite groups up to order 36".