Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. The smallest non-abelian group has 6 elements. It is a dihedral group with notation D3 (or D6, both are used) and the symmetric group of degree 3, with notation S3. This page illustrates many group concepts using this group as example. In 2D the group D3 is the symmetry group of an equilateral triangle. As opposed to the case of e.g. a square, all permutations of the vertices can be achieved by rotation and flipping over (or reflecting). In 3D there are two different symmetry groups which are algebraically the group D3: one with a 3-fold rotation axis and a perpendicular 2-fold rotation axis (hence three of these): D3. One with a 3-fold rotation axis in a plane of reflection (and hence also in two other planes of reflection): C3v.

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