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In mathematics, D 3 (sometimes alternatively denoted by D 6) is the dihedral group of degree 3 and order 6. It equals the symmetric group S 3. It is also the smallest non-abelian group. [1] This page illustrates many group concepts using this group as example.
List of all nonabelian groups up to order 31 Order Id. [a] G o i Group Non-trivial proper subgroups [1] Cycle graph Properties 6 7 G 6 1: D 6 = S 3 = Z 3 ⋊ Z 2: Z 3, Z 2 (3) : Dihedral group, Dih 3, the smallest non-abelian group, symmetric group, smallest Frobenius group.
In mathematics, a dihedral group is the group of symmetries of a regular polygon, [1] [2] which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, geometry, and chemistry. [3] The notation for the dihedral group differs in geometry and abstract ...
The group order is defined as the subscript, unless the order is doubled for symbols with a plus or minus, "±", prefix, which implies a central inversion. [ 3 ] Hermann–Mauguin notation (International notation) is also given.
There are 16 subgroups. There are 8 up to isomorphism: itself (D 6), 2 dihedral: (D 3, D 2), 4 cyclic: (Z 6, Z 3, Z 2, Z 1) and the trivial (e) These symmetries express nine distinct symmetries of a regular hexagon. John Conway labels these by a letter and group order. [4] r12 is full symmetry, and a1 is no symmetry.
In geometry, many uniform tilings on sphere, euclidean plane, and hyperbolic plane can be made by Wythoff construction within a fundamental triangle, (p q r), defined by internal angles as π/p, π/q, and π/r.
The smallest abstract groups that are not any symmetry group in 3D, are the quaternion group (of order 8), Z 3 × Z 3 (of order 9), the dicyclic group Dic 3 (of order 12), and 10 of the 14 groups of order 16. The column "# of order 2 elements" in the following tables shows the total number of isometry subgroups of types C 2, C i, C s. This ...
D 1d and C 2h: group of order 4 with a reflection in a plane and a 180° rotation about a line perpendicular to that plane. For n = 2 there is not one main axis and two additional axes, but there are three equivalent ones. D 2, [2,2] +, (222) of order 4 is one of the three symmetry group types with the Klein four-group as abstract group. It has ...