Encyclopedia > List of small groups

  Article Content

List of small groups

This is a list of small finite mathematical groups. For each order, all groups of that order up to group isomorphism are listed.

Glossary

The notation G × H stands for the direct product of the two groups.

List

Order Groups
1 C1 (the trivial group, abelian)
2 C2 (abelian, simple)
3 C3 (abelian, simple)
4 C4 (abelian); C2 × C2 (abelian, isomorphic to the Klein four-group).
5 C5 (abelian, simple)
6 C6 (abelian); S3 (isomorphic to D6, the smallest non-abelian group)
7 C7 (abelian, simple)
8 C8 (abelian); C2 × C4 (abelian); C2 × C2 × C2 (abelian); D8; Q8 (the quaternion group)
9 C9 (abelian); C3 × C3 (abelian)
10 C10 (abelian); D10
11 C11 (abelian, simple)
12 C12 (abelian); C2 × C6 (abelian); D12; A4; the semidirect product of C3 and C4, where C4 acts on C3 by inversion.
13 C13 (abelian, simple)
14 C14 (abelian); D14
15 C15 (abelian)

Please add higher orders, and/or more information about the groups (maximal subgroups, normal subgroups, character tables etc.)


The group theoretical computer algebra system GAP (available for free at http://www.gap-system.org/ ) contains the "Small Groups library": it 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 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 .



All Wikipedia text is available under the terms of the GNU Free Documentation License

 
  Search Encyclopedia

Search over one million articles, find something about almost anything!
 
 
  
  Featured Article
Johann Karl Friedrich Rosenkranz

... his last years he was quite blind. Throughout his long professorial career, and in all his numerous publications he remained, in spite of occasional deviations on ...

 
 
 
This page was created in 38.7 ms