Groups of order < 30


There is an excellent webpage with a much larger database at http://u.cs.biu.ac.il/~mschaps/DATA/database.html. (If that is down, try here.)

The following page was created by Prof Pedersen, and "orphaned." Years ago, I was given permission to post and maintain it here. Feel free to use it however you like but not Prof Schaps' page is much more extensive.


The mathematical software programs GAP and MAGMA have a (common) SmallGroups database for small groups. This web page illustrates the cases of groups of order < 30.

All groups of prime order p are isomorphic to Cp, the cyclic group of order p and are omitted. A concrete realization of this group is ZZ/pZZ, the integers under addition modulo p.

Order 1 and all prime orders (1 group: 1 abelian, 0 nonabelian)

Order 4 (2 groups: 2 abelian, 0 nonabelian)


Order 6 (2 groups: 1 abelian, 1 nonabelian)


Order 8 (5 groups: 3 abelian, 2 nonabelian)


Order 9 (2 groups: 2 abelian, 0 nonabelian)


Order 10 (2 groups: 1 abelian, 1 nonabelian)


Order 12 (5 groups: 2 abelian, 3 nonabelian)


Order 14 (2 groups: 1 abelian, 1 nonabelian)


Order 15 (1 group: 1 abelian, 0 nonabelian)

C15

Order 16 (14 groups: 5 abelian, 9 nonabelian)


Order 18 (5 groups: 2 abelian, 3 nonabelian)


Order 20 (5 groups: 2 abelian, 3 nonabelian)


Order 21 (2 groups: 1 abelian, 1 nonabelian)


Order 22 (2 groups: 1 abelian, 1 nonabelian)


Order 24 (15 groups: 3 abelian, 12 nonabelian)


Order 25 (2 groups: 2 abelian, 0 nonabelian)


Order 26 (2 groups: 1 abelian, 1 nonabelian)


Order 27 (5 groups: 3 abelian, 2 nonabelian)


Order 28 (4 groups: 2 abelian, 2 nonabelian)


Order 30 (4 groups: 1 abelian, 3 nonabelian)


Compiled by John Pedersen and David Joyner (wdjoyner@gmail.com).
Distributed under the Attribution-ShareAlike Creative Commons license.
Last updated 2012-04-26.