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S 3 S 3 is the first nonabelian symmetric group. This group is isomorphic to the dihedral group of order 6, the group of reflection and rotation symmetries of an equilateral triangle, since these symmetries permute the three vertices of the triangle. Cycles of length two correspond to reflections, and cycles of length three are rotations.
For n = 3, 4 there are two additional one-dimensional irreducible representations, corresponding to maps to the cyclic group of order 3: A 3 ≅ C 3 and A 4 → A 4 /V ≅ C 3. For n ≥ 7, there is just one irreducible representation of degree n − 1, and this is the smallest degree of a non-trivial irreducible representation.
[2] [3] The specific nature of the outer automorphism is as follows. The 360 permutations in the even subgroup (A 6) are transformed amongst themselves: the sole identity permutation maps to itself; a 3-cycle such as (1 2 3) maps to the product of two 3-cycles such as (1 4 5)(2 6 3) and vice versa, accounting for 40 permutations each way;
In three dimensions, the hyperoctahedral group is known as O × S 2 where O ≅ S 4 is the octahedral group, and S 2 is a symmetric group (here a cyclic group) of order 2. Geometric figures in three dimensions with this symmetry group are said to have octahedral symmetry, named after the regular octahedron, or 3-orthoplex.
In all these cases except for D 4, there is a single non-trivial automorphism (Out = C 2, the cyclic group of order 2), while for D 4, the automorphism group is the symmetric group on three letters (S 3, order 6) – this phenomenon is known as "triality". It happens that all these diagram automorphisms can be realized as Euclidean symmetries ...
When comparing the symmetry type of two objects, the origin is chosen for each separately, i.e., they need not have the same center. Moreover, two objects are considered to be of the same symmetry type if their symmetry groups are conjugate subgroups of O(3) (two subgroups H 1, H 2 of a group G are conjugate, if there exists g ∈ G such that H 1 = g −1 H 2 g).
The group (,,) is the generalized symmetric group: algebraically, it is the wreath product (/) of the cyclic group / with the symmetric group . Concretely, the elements of the group may be represented by monomial matrices (matrices having one nonzero entry in every row and column) whose nonzero entries are all m th roots of unity.
In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the ambient space which takes the object to itself, and which preserves all the relevant structure of the object.