List of small groups
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For the number of nonisomorphic groups of order is
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:
- Zn: the cyclic group of order n (the notation Cn is also used; it is isomorphic to the additive group of Z/nZ).
- Dihn: the dihedral group of order 2n (often the notation Dn or D2n is used )
- K4: the Klein four-group of order 4, same as Z2 × Z2 or Dih2.
- Sn: the symmetric group of degree n, containing the n! permutations of n elements.
- An: 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.
- Dicn or Q4n: the dicyclic group of order 4n.
- Q8: the quaternion group of order 8, also Dic2.
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. G ⋊ H denotes a semidirect product where H acts on G; this may also depend on the choice of action of H on G
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.
List of small abelian groupsEdit
The finite abelian groups are either cyclic groups, or direct products thereof; see abelian groups. The numbers of nonisomorphic abelian groups of orders are
|Order||ID||Goi||Group||Nontrivial proper Subgroups||Cycle
|1||1||G11||Z1 = S1 = A2||–||Trivial. Cyclic. Alternating. Symmetric. Elementary.|
|2||2||G21||Z2 = S2 = Dih1||–||Simple. Symmetric. Cyclic. Elementary. (Smallest non-trivial group.)|
|3||3||G31||Z3 = A3||–||Simple. Alternating. Cyclic. Elementary.|
|4||4||G41||Z4 = Dic1||Z2||Cyclic.|
|5||G42||Z22 = K4 = Dih2||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||Z3, Z2||Cyclic. Product.|
|7||9||G71||Z7||–||Simple. Cyclic. Elementary.|
|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.)|
|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||Z6, Z3, Z2||Product.|
|19||49||G191||Z19||–||Simple. Cyclic. Elementary.|
|20||51||G202||Z20 = Z5 × Z4||Z20, Z10, Z5, Z4, Z2||Cyclic. Product.|
|54||G205||Z10 × Z2 = Z5 × Z22||Z5, Z2||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.|
|26||78||G262||Z26 = Z13 × Z2||Z13, Z2||Cyclic. Product.|
|80||G272||Z9 × Z3||Z9, Z3||Product.|
|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 groupsEdit
- 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)
|Order||ID||Goi||Group||Nontrivial proper Subgroups||Cycle
|6||7||G61||Dih3 = S3 = D6||Z3, Z2 (3)||Dihedral group, the smallest non-abelian group, symmetric group, Frobenius group|
|8||12||G83||Dih4 = D8||Z4, Z22 (2), Z2 (5)||Dihedral group. Extraspecial group. Nilpotent.|
|13||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||17||G101||Dih5 = D10||Z5, Z2 (5)||Dihedral group, Frobenius group|
|12||20||G121||Q12 = Dic3 = <3,2,2>
= Z3 ⋊ Z4
|Z2, Z3, Z4 (3), Z6||Binary dihedral group|
|22||G123||A4||Z22, Z3 (4), Z2 (3)||Alternating group. No subgroups of order 6, although 6 divides its order. Frobenius group|
|23||G124||Dih6 = D12 = Dih3 × Z2||Z6, Dih3 (2), Z22 (3), Z3, Z2 (7)||Dihedral group, product|
|14||26||G141||Dih7 = D14||Z7, Z2 (7)||Dihedral group, Frobenius group|
|16||31||G163||G4,4 = K4 ⋊ Z4
(Z4×Z2) ⋊ Z2
|E8, Z4 × Z2 (2), Z4 (4), K4 (6), Z2 (6)||Has the same number of elements of every order as the Pauli group. Nilpotent.|
|32||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.|
|34||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.|
|35||G167||Dih8 = D16||Z8, Dih4 (2), Z22 (4), Z4, Z2 (9)||Dihedral group. Nilpotent.|
|36||G168||QD16||The order 16 quasidihedral group. Nilpotent.|
|37||G169||Q16 = Dic4 = <4,2,2>||generalized quaternion group, binary dihedral group. Nilpotent.|
|39||G1611||Dih4 × Z2||Dih4 (4), Z4 × Z2, Z23 (2), Z22 (13), Z4 (2), Z2 (11)||Product. Nilpotent.|
|40||G1612||Q8 × Z2||Hamiltonian, product. Nilpotent.|
|41||G1613||(Z4 × Z2) ⋊ Z2||The Pauli group generated by the Pauli matrices. Nilpotent.|
|18||44||G181||Dih9 = D18||Dihedral group, Frobenius group|
|46||G183||S3 × Z3||Product|
|47||G184||(Z3 × Z3) ⋊ Z2||Frobenius group|
|20||50||G201||Q20 = Dic5 = <5,2,2>||Binary dihedral group|
|52||G203||Z5 ⋊ Z4||Frobenius group|
|53||G204||Dih10 = Dih5 × Z2 = D20||Dihedral group, product|
|21||55||G211||Z7 ⋊ Z3||Z7, Z3 (7)||Smallest non-abelian group of odd order. Frobenius group|
|22||57||G221||Dih11 = D22||Z11, Z2 (11)||Dihedral group, Frobenius group|
|24||60||G241||Z3 ⋊ Z8||Central extension of S3|
|62||G243||SL(2,3) = 2T = Q8 ⋊ Z3||Binary tetrahedral group|
|63||G244||Q24 = Dic6 = <6,2,2> = Z3 ⋊ Q8||Binary dihedral|
|64||G245||Z4 × S3||Product|
|66||G247||Dic3 × Z2 = Z2 × (Z3 ⋊ Z4)||Product|
|67||G248||(Z6 × Z2) ⋊ Z2 = Z3 ⋊ Dih4||Double cover of dihedral group|
|69||G2410||Dih4 × Z3||Product. Nilpotent.|
|70||G2411||Q8 × Z3||Product. Nilpotent.|
|71||G2412||S4||Symmetric group. Has no normal Sylow subgroups.|
|72||G2413||A4 × Z2||Product|
|26||77||G261||Dih13||Dihedral group, Frobenius group|
|27||81||G273||Z32 ⋊ Z3||All non-trivial elements have order 3. Extraspecial group. Nilpotent.|
|82||G274||Z9 ⋊ Z3||Extraspecial group. Nilpotent.|
|28||84||G281||Z7 ⋊ Z4||Binary dihedral group|
|86||G283||Dih14||Dihedral group, product|
|30||89||G301||Z5 × S3||Product|
|90||G302||Z3 × Dih5||Product|
|91||G303||Dih15||Dihedral group, Frobenius group|
Classifying groups of small orderEdit
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.
Small groups libraryEdit
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:
- those of order at most 2000;
- 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 SmallGroups library does not have information is 2048.
- 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)". 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.