Dihedral group of order 6

In mathematics, D3 (sometimes alternatively denoted by D6) is the dihedral group of degree 3 and order 6.

The dihedral group D3 is the symmetry group of an equilateral triangle, that is, it is the set of all rigid transformations (reflections, rotations, and combinations of these) that leave the shape and position of this triangle fixed.

In the case of D3, every possible permutation of the triangle's vertices constitutes such a transformation, so that the group of these symmetries is isomorphic to the symmetric group S3 of all permutations of three distinct elements.

This is not the case for dihedral groups of higher orders.

The dihedral group D3 is isomorphic to two other symmetry groups in three dimensions: Consider three colored blocks (red, green, and blue), initially placed in the order RGB.

In multiplicative form, we traditionally write xy for the combined action "first do y, then do x"; so that ab is the action RGB ↦ RBG ↦ BRG, i.e., "take the last block and move it to the front".

If we write e for "leave the blocks as they are" (the identity action), then we can write the six permutations of the set of three blocks as the following actions: The notation in parentheses is the cycle notation.

Note that the action aa has the effect RGB ↦ GRB ↦ RGB, leaving the blocks as they were; so we can write aa = e. Similarly, so each of the above actions has an inverse.

The group has presentation where a and b are swaps and r = ab is a cyclic permutation.

With the generators a and b, we define the additional shorthands c := aba, d := ab and f := ba, so that a, b, c, d, e, and f are all the elements of this group.

We can then summarize the group operations in the form of a Cayley table: Note that non-equal non-identity elements only commute if they are each other's inverse.

We can easily distinguish three kinds of permutations of the three blocks, the conjugacy classes of the group: For example, (RG) and (RB) are both of the form (x y); a permutation of the letters R, G, and B (namely (GB)) changes the notation (RG) into (RB).

From Lagrange's theorem we know that any non-trivial subgroup of a group with 6 elements must have order 2 or 3.

In fact the two cyclic permutations of all three blocks, with the identity, form a subgroup of order 3, index 2, and the swaps of two blocks, each with the identity, form three subgroups of order 2, index 3.

The existence of subgroups of order 2 and 3 is also a consequence of Cauchy's theorem.

If the original group is that generated by a 120°-rotation of a plane about a point, and reflection with respect to a line through that point, then the quotient group has the two elements which can be described as the subsets "just rotate (or do nothing)" and "take a mirror image".

Note that for the symmetry group of a square, an uneven permutation of vertices does not correspond to taking a mirror image, but to operations not allowed for rectangles, i.e. 90° rotation and applying a diagonal axis of reflection.

The semidirect product is isomorphic to the dihedral group of order 6 if φ(0) is the identity and φ(1) is the non-trivial automorphism of C3, which inverses the elements.

Consider D3 in the geometrical way, as a symmetry group of isometries of the plane, and consider the corresponding group action on a set of 30 evenly spaced points on a circle, numbered 0 to 29, with 0 at one of the reflexion axes.

This section illustrates group action concepts for this case.

If a symmetry group applies for a pattern, then within each orbit the color is the same.

The image of this map is the orbit of x and the coimage is the set of all left cosets of Gx.

If two elements x and y belong to the same orbit, then their stabilizer subgroups, Gx and Gy, are isomorphic.

A result closely related to the orbit-stabilizer theorem is Burnside's lemma: where Xg is the set of points fixed by g. I.e., the number of orbits is equal to the average number of points fixed per group element.

Up to isomorphism, this group has three irreducible complex unitary representations, which we will call

grading: The action is multiplication by the sign of the permutation of the group element.

[2] A 2-dimensional irreducible linear representation yields a 1-dimensional projective representation (i.e., an action on the projective line, an embedding in the Möbius group PGL(2, C)), as elliptic transforms.

This can be represented by matrices with entries 0 and ±1 (here written as fractional linear transformations), known as the anharmonic group: and thus descends to a representation over any field, which is always faithful/injective (since no two terms differ only by only a sign).

Over the field with two elements, the projective line has only 3 points, and this is thus the exceptional isomorphism

(in characteristic greater than 3 these points are distinct and permuted, and are the orbit of the harmonic cross-ratio).