groups of small order

Below is a list of all possible groups per order up to isomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath.

Groups of prime order:

  • All groups of prime order are isomorphic to a cyclic groupMathworldPlanetmath of that order.

Groups of prime square order:

  • All groups of order p2, where p is a prime, are isomorphic to one of the following:

Groups of order 1:

Groups of order 6:

Groups of order 8:

  • C8(Abelian): cyclic group of order 8.

  • C4×C2(Abelian): direct productMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath of two groups of a cyclic group of order 4 and a cyclic group of order 2.

  • C2×C2×C2(Abelian): direct product of three groups of a cyclic group of order 2.

  • D4(non-Abelian): octic group; dihedral groupMathworldPlanetmath of degree 4.

  • Q8(non-Abelian): quaternion groupMathworldPlanetmathPlanetmath.

Groups of order 10:

  • C10(Abelian): cyclic group of order 10.

  • D5(non-Abelian): dihedral group of degree 5.

Groups of order 12:

  • C12(Abelian): cyclic group of order 12.

  • C2×C6(Abelian).

  • A4(non-Abelian): alternating groupMathworldPlanetmath of degree 4.

  • D6(non-Abelian): dihedral group of degree 6.

  • Dic(C6)(non-Abelian): dicyclic group of order 12. This is a generalized quaternion group Q12.

Groups of order 14:

  • C14(Abelian): cyclic group of order 14.

  • D7(non-Abelian): dihedral group of degree 7.

Groups of order 15:

  • C15(Abelian): cyclic group of order 15.


  • PJ Pedersen, John: Groups of small order. eclark/algctlg/small_groups.html eclark/algctlg/small_groups.html
Title groups of small order
Canonical name GroupsOfSmallOrder
Date of creation 2013-03-22 14:47:54
Last modified on 2013-03-22 14:47:54
Owner Daume (40)
Last modified by Daume (40)
Numerical id 15
Author Daume (40)
Entry type Example
Classification msc 20A05
Classification msc 20-00
Related topic ExamplesOfGroups