# groups of small order

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

Groups of prime order:

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

Groups of prime square order:

• All groups of order $p^{2}$, where $p$ is a prime, are isomorphic to one of the following:

• $C_{p^{2}}$(Abelian): cyclic group of order $p^{2}$.

• $C_{p}\times C_{p}$(Abelian): elementary abelian group of order $p^{2}$.

Groups of order 1:

• trivial group (i.e. $\{e\}$).

Groups of order 6:

• $C_{6}$(Abelian): cyclic group of order 6.

• $S_{3}$: symmetric group where $n=3$.

Groups of order 8:

• $C_{8}$(Abelian): cyclic group of order 8.

• $C_{4}\times C_{2}$(Abelian): direct product of two groups of a cyclic group of order 4 and a cyclic group of order 2.

• $C_{2}\times C_{2}\times C_{2}$(Abelian): direct product of three groups of a cyclic group of order 2.

• $D_{4}$(non-Abelian): octic group; dihedral group of degree 4.

• $Q_{8}$(non-Abelian): quaternion group.

Groups of order 10:

• $C_{10}$(Abelian): cyclic group of order 10.

• $D_{5}$(non-Abelian): dihedral group of degree 5.

Groups of order 12:

• $C_{12}$(Abelian): cyclic group of order 12.

• $C_{2}\times C_{6}$(Abelian).

• $A_{4}$(non-Abelian): alternating group of degree 4.

• $D_{6}$(non-Abelian): dihedral group of degree 6.

• $\operatorname{Dic}(C_{6})$(non-Abelian): dicyclic group of order 12. This is a generalized quaternion group $Q_{12}$.

Groups of order 14:

• $C_{14}$(Abelian): cyclic group of order 14.

• $D_{7}$(non-Abelian): dihedral group of degree 7.

Groups of order 15:

• $C_{15}$(Abelian): cyclic group of order 15.

## References

• PJ Pedersen, John: Groups of small order. http://www.math.usf.edu/ eclark/algctlg/small_groups.htmlhttp://www.math.usf.edu/ eclark/algctlg/small_groups.html
Title groups of small order GroupsOfSmallOrder 2013-03-22 14:47:54 2013-03-22 14:47:54 Daume (40) Daume (40) 15 Daume (40) Example msc 20A05 msc 20-00 ExamplesOfGroups