List of small groups
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.
- 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 )
- Sn: the symmetric group of degree n, containing the n! permutations of n elements.
- An: the alternating group of degree n, containing the n!/2 even permutations of n elements.
- Dicn: the dicyclic group of order 4n.
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 stands for the direct product of the two groups; Gn denotes the direct product of a group with itself n times. G ⋊ H stands for a semidirect product where H acts on G; where the particular action of H on G is omitted, it is because all possible non-trivial actions result in the same product group, up to 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.
|1||trivial group, Z1 = S1 = A2||–||various properties hold trivially|
|2||Z2 = S2 = Dih1||–||simple, the smallest non-trivial group|
|3||Z3 = A3||–||simple|
|Klein four-group, Z 2
2 = Dih2
|Z2 (3)||the smallest non-cyclic group|
|6||Z6 = Z3 × Z2||Z3, Z2|
|Z4 × Z2||Z 2
2 , Z4 (2), Z2 (3)
2 (7), Z2 (7)
|the non-identity elements correspond to the points in the Fano plane, the Z2 × Z2 subgroups to the lines|
|10||Z10 = Z5 × Z2||Z5, Z2|
|12||Z12 = Z4 × Z3||Z6, Z4, Z3, Z2|
|Z6 × Z2 = Z3 × Z 2
|Z6 (3), Z3, Z2 (3), Z 2
|14||Z14 = Z7 × Z2||Z7, Z2|
|15||Z15 = Z5 × Z3||Z5, Z3||multiplication of nimbers 1,...,15|
|16||Z16||Z8, Z4, Z2|
|Z2 (15), Z 2
2 (35), Z 3
|addition of nimbers 0,...,15|
|Z4 × Z 2
|Z2 (7), Z4 (4), Z 2
2 (7), Z 3
2 , Z4 × Z2 (6)
|Z8 × Z2||Z2 (3), Z4 (2), Z 2
2 , Z8 (2), Z4 × Z2
|Z2 (3), Z4 (6), Z 2
2 , Z4 × Z2 (3)
List of small non-abelian groups
|6||S3 = Dih3||Z3, Z2 (3)||the smallest non-abelian group|
|8||Z4, Z22 (2), Z2 (5)|
|quaternion group, Q8 = Dic2||Z4 (3), Z2||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||Dih5||Z5, Z2 (5)|
|12||Dih6 = Dih3 × Z2||Z6, Dih3 (2), Z22 (3), Z3, Z2 (7)|
|A4||Z22, Z3 (4), Z2 (3)||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.)|
|Dic3 = Z3 ⋊ Z4||Z2, Z3, Z4 (3), Z6|
|14||Dih7||Z7, Z2 (7)|
|16||Dih8||Z8, Dih4 (2), Z22 (4), Z4, Z2 (9)|
|Dih4 × Z2||Dih4 (2), Z4 × Z2, Z23 (2), Z22 (11), Z4 (2), Z2 (11)|
|generalized quaternion group, Q16 = Dic4|
|Q8 × Z2||Hamiltonian|
|The order 16 quasidihedral group|
|The order 16 modular group|
|Z4 ⋊ Z4|
|The group generated by the Pauli matrices|
|G4,4 = Z22 ⋊ Z4|
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:
- 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.
- Classification of finite simple groups, the four categories
- Composition series, some groups may be broken down into simpler groups
- List of finite simple groups
- Number of groups of a given order
- Small Latin squares and quasigroups
- See a worked example showing the isomorphism Z6 = Z3 × Z2.
- Wild, Marcel. "The Groups of Order Sixteen Made Easy", American Mathematical Monthly, Jan 2005
- Hans Ulrich Besche The Small Groups library
- Particular groups groupprops.subwiki.org
- Besche, H. U.; Eick, B.; O'Brien, E. "small group library".
- Hall, Jr., Marshall; Senior, James K. (1964). The Groups of Order 2n (n ≤ 6). Macmillan. LCCN 64016861. MR 168631. An exhaustive catalog of the 340 groups of order dividing 64 with detailed tables of defining relations, constants, and lattice presentations of each group in the notation the text defines. "Of enduring value to those interested in finite groups" (from the preface).