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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;
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 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 ...
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.
The symmetric group of degree n ≥ 4 has Schur covers of order 2⋅n! There are two isomorphism classes if n ≠ 6 and one isomorphism class if n = 6. The alternating group of degree n has one isomorphism class of Schur cover, which has order n! except when n is 6 or 7, in which case the Schur cover has order 3⋅n!.
The first group homology group (concretely, the abelianization) is (for m odd this is isomorphic to ): the factors (which are all conjugate, hence must map identically in an abelian group, since conjugation is trivial in an abelian group) can be mapped to (concretely, by taking the product of all the values), while the sign map on the symmetric group yields the .