Groups of small order

Compiled by John Pedersen, Dept of Mathematics, University of South Florida

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

All groups of prime order p are isomorphic to C_p, the cyclic group of order p.
A concrete realization of this group is Z_p, the integers under addition modulo p.

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)

C_15.

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)


A Catalogue of Algebraic Systems / John Pedersen / jfp@math.usf.edu