Science Fair Project Encyclopedia
List of small groups
The following list in mathematics contains the finite groups of small order up to group isomorphism.
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.
Glossary
- Cn: the cyclic group of order n.
- Dn: the dihedral group of order n.
- 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 notation G × H stands for the direct product of the two groups. Abelian and simple groups are noted. (For groups of order n < 60, the simple groups are precisely the cyclic groups Cn, where n is prime.) We use the equality sign ("=") to denote isomorphism.
The identity element in the cycle graphs are represented by the black circle.
List
| Order | Group | Properties | Cycle graph |
|---|---|---|---|
| 1 | trivial group = C1 = S1 = A2 | abelian | |
| 2 | C2 = S2 | abelian, simple, the smallest non-trivial group | |
| 3 | C3 = A3 | abelian, simple | |
| 4 | C4 | abelian, | |
| Klein four-group = C2 × C2 = D4 | abelian, the smallest non-cyclic group | ||
| 5 | C5 | abelian, simple | |
| 6 | C6 = C2 × C3 | abelian | |
| S3 = D6 | the smallest non-abelian group | ||
| 7 | C7 | abelian, simple | |
| 8 | C8 | abelian | |
| C2 ×C4 | abelian | ||
| C2 × C2 × C2 = D4 × C2 | abelian | ||
| D8 | non-abelian | ||
| Quaternion group, Q8 = Dic2 | non-abelian; the smallest Hamiltonian group | ||
| 9 | C9 | abelian | |
| C3 × C3 | abelian | ||
| 10 | C10 = C2 × C5 | abelian | |
| D10 | non-abelian | ||
| 11 | C11 | abelian, simple | |
| 12 | C12 = C4 × C3 | abelian | |
| C2 × C6 = C2 × C2 × C3 = D4 × C3 | abelian | ||
| D12 = D6 × C2 | non-abelian | ||
| A4 | non-abelian | ||
| Dic3 = the semidirect product of C3 and C4, where C4 acts on C3 by inversion | non-abelian | ||
| 13 | C13 | abelian, simple | |
| 14 | C14 = C2 × C7 | abelian | |
| D14 | non-abelian | ||
| 15 | C15 = C3 × C5 | abelian | |
| 16 | C16 | abelian | |
| C2 × C2 × C2 × C2 | abelian | ||
| C2 × C2 × C4 | abelian | ||
| C2 × C8 | abelian | ||
| C4 × C4 | abelian | ||
| D16 | non-abelian | ||
| Generalized quaternion group, Q16 = Dic4 | non-abelian | ||
| C2 × D8 | non-abelian | ||
| C2 × Q8 | non-abelian | ||
| The order 16 quasidihedral group | non-abelian | ||
| The order 16 modular group | non-abelian | ||
| The semidirect product of C4 and C4 where one factor acts on the other by inversion | non-abelian | ||
| The group generated by the Pauli matrices | non-abelian | ||
| G4,4 | non-abelian |
- Please add higher orders, and/or more information about the groups (maximal subgroups, normal subgroups, character tables etc.)
Small groups library
The group theoretical computer algebra system GAP contains the "Small Groups library" which provides access to descriptions of the groups of "small" order. 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 (423 164 062 groups);
- those of order 5^5 and 7^4 (92 groups);
- those of order q^n * p where q^n divides 2^8, 3^6, 5^5 or 7^4 and p is an arbitrary prime which differs from q;
- those whose order factorises into at most 3 primes.
It contains explicit descriptions of the available groups in computer readable format.
The library has been constructed and prepared by Hans Ulrich Besche, Bettina Eick and Eamonn O'Brien; see http://www.tu-bs.de/~hubesche/small.html .
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