List of small groups

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

Counts

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

Each group is named by Small Groups library as G_oⁱ, where o is the order of the group, and i is the index used to label 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)
• Dih_n: the dihedral group of order 2n (often the notation D_n or D_2n is used)
  • K₄: the Klein four-group of order 4, same as Z₂ × Z₂ and Dih₂
• D_2n: the dihedral group of order 2n, the same as Dih_n (notation used in section List of small non-abelian groups)
• 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 even permutations of n elements, of order 1 for n = 0, 1, and order n!/2 otherwise
• 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 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

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. [lower-alpha 1]	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 = Dic 1	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		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

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. [lower-alpha 1]	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

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

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

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

Notes

1. Identifier when groups are numbered by order, o, then by index, i, from the small groups library, starting at 1.

1. 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. 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>: R^l=S^m=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

• 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.

External links

• Particular groups in the Group Properties Wiki
• Besche, H.U.; Eick, B.; O'Brien, E. "Small Group Library". Archived from the original on 2012-03-05.
• GroupNames database
• Hall, Jr., Marshall; Senior, James Kuhn (1964). The Groups of Order 2ⁿ (n ≤ 6). New York: Macmillan / London: Collier-Macmillan Ltd. LCCN 64016861