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In mathematics, 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.〔. For the identification of D3 with S3, and the observation that this group is the smallest possible non-abelian group, see (p. 49 ).〕 This page illustrates many group concepts using this group as example. ==Symmetry groups== In two dimensions, the group D3 is the symmetry group of an equilateral triangle. In contrast with the case of a square or other polygon, all permutations of the vertices can be achieved by rotation and flipping over (or reflecting). :160px In three dimensions, 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 :160px 160px 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Dihedral group of order 6」の詳細全文を読む スポンサード リンク
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