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.
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 Group s than these two, of the same abstract group type.
The notation ''G'' × ''H'' stands for the Direct Product of the two groups. Abelian and Simple Group s are noted. (For groups of order ''n'' < 60, the simple groups are precisely the cyclic groups Z''n'', 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. 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.
See also the List Of Small Abelian Groups and the combined list below.
Note that e.g. "3 × Z2" means that there are 3 subgroups of type Z2 (NOT a left coset of Z2), while elsewhere the cross means Direct Product .
Order |
Group |
Subgroups |
Properties |
Cycle graph |
|---|
6 |
''S''3 = Dih3 |
Z3 , 3 × Z2 |
the smallest non-abelian group |
|
|---|
8 |
Dih4 |
Z4, 2 × Dih2 , 5 × Z2 |
non-abelian |
|
|---|
Quaternion Group , ''Q''8 = Dic2 |
3 × Z4 , Z2 |
non-abelian; the smallest Hamiltonian Group |
|
10 |
Dih5 |
Z5 , 5 × Z2 |
non-abelian |
|
|---|
12 |
|---|
14 |
Dih7 |
Z7 , 7 × Z2 |
non-abelian |
|
|---|
16 |
Order |
Group |
Subgroups |
Properties |
Cycle graph |
|---|
1 |
= Z1 = ''S''1 = ''A''2 |
- |
abelian; this and various other properties hold Trivially
|
|
|---|
2 |
Z2 = ''S''2 = Dih1 |
- |
abelian, simple, the smallest non-trivial group |
|
|---|
3 |
Z3 = ''A''3 |
- |
abelian, simple |
|
|---|
4 |
Z4 |
Z2 |
abelian |
|
|---|
Klein Four-group = Z2 ×
Z2 = Dih2 |
3 × Z2 |
abelian, the smallest non-cyclic group |
|
5 |
Z5 |
- |
abelian, simple |
|
|---|
6 |
Z6 = Z2 × Z3 |
Z2 , Z3 |
abelian |
|
|---|
''S''3 = Dih3 |
Z3 , 3 × Z2 |
the smallest non-abelian group |
|
7 |
Z7 |
- |
abelian, simple |
|
|---|
8 |
Z8 |
Z4 , Z2 |
abelian |
|
|---|
Z2 ×Z4 |
2 × Z4 , 3 ×Z2 , Dih2 |
abelian |
|
Z2 ×
Z2 × Z2 = Dih2 × Z2 |
7 ×
Z2 × Z2 , 7 × Z2 |
abelian |
|
Dih4 |
Z4, 2 × Dih2 , 5 × Z2 |
non-abelian |
|
Quaternion Group , ''Q''8 = Dic2 |
3 × Z4 , Z2 |
non-abelian; the smallest Hamiltonian Group |
|
9 |
Z9 |
Z3 |
abelian |
|
|---|
Z3 ×
Z3 |
4 × Z3 |
abelian |
|
10 |
Z10 = Z2 × Z5 |
Z5 , Z2 |
abelian |
|
|---|
Dih5 |
Z5 , 5 × Z2 |
non-abelian |
|
11 |
Z11 |
- |
abelian, simple |
|
|---|
12 |
Z12 = Z4 × Z3 |
Z6 , Z4 , Z3 , Z2 |
abelian |
|
|---|
Z2 × Z6 = Z2 ×
Z2 × Z3 = Dih2 × Z3 |
3 × Z6, Z3, Dih2, 3 × Z2
| abelian |
|
|
13 |
Z13 |
- |
abelian, simple |
|
|---|
14 |
Z14 = Z2 × Z7 |
Z7 , Z2 |
abelian |
|
|---|
Dih7 |
Z7 , 7 × Z2 |
non-abelian |
|
15 |
Z15 = Z3 × Z5 |
Z5 , Z3 |
abelian |
|
|---|
16 |
|---|
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 55 and 74 (92 groups);
- those of order ''q''''n''×''p'' where ''q''''n'' divides 28, 36, 55 or 74 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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