groups of small order
Below is a list of all possible groups per order up to isomorphism^{}.
Groups of prime order:

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All groups of prime order are isomorphic to a cyclic group^{} of that order.
Groups of prime square order:

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

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${C}_{{p}^{2}}$(Abelian^{}): cyclic group of order ${p}^{2}$.

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

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Groups of order 1:

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trivial group (i.e. $\mathrm{\{}e\mathrm{\}}$).
Groups of order 6:

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${C}_{6}$(Abelian): cyclic group of order 6.

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${S}_{3}$(nonAbelian^{}): symmetric group^{} where $n=3$.
Groups of order 8:

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${C}_{8}$(Abelian): cyclic group of order 8.

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

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

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${D}_{4}$(nonAbelian): octic group; dihedral group^{} of degree 4.

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${Q}_{8}$(nonAbelian): quaternion group^{}.
Groups of order 10:

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${C}_{10}$(Abelian): cyclic group of order 10.

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${D}_{5}$(nonAbelian): dihedral group of degree 5.
Groups of order 12:

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${C}_{12}$(Abelian): cyclic group of order 12.

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${C}_{2}\times {C}_{6}$(Abelian).

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${A}_{4}$(nonAbelian): alternating group^{} of degree 4.

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${D}_{6}$(nonAbelian): dihedral group of degree 6.

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$\mathrm{Dic}({C}_{6})$(nonAbelian): dicyclic group of order 12. This is a generalized quaternion group ${Q}_{12}$.
Groups of order 14:

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${C}_{14}$(Abelian): cyclic group of order 14.

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${D}_{7}$(nonAbelian): dihedral group of degree 7.
Groups of order 15:

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${C}_{15}$(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 

Canonical name  GroupsOfSmallOrder 
Date of creation  20130322 14:47:54 
Last modified on  20130322 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 2000 
Related topic  ExamplesOfGroups 