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A Cayley graph of the symmetric group S 4 using the generators (red) a right circular shift of all four set elements, and (blue) a left circular shift of the first three set elements. Cayley table, with header omitted, of the symmetric group S 3. The elements are represented as matrices. To the left of the matrices, are their two-line form.
Dih 4: Dihedral group of order 8 S 3: Symmetric group of order 6 C 2 2: Klein 4-group C 4: Cyclic group of order 4 C 3: 3 element group C 2: 2 element group C 1: Trivial group. The C 2 knots are marked with the index numbers (compare this table) of the transpositions (1, 6, 5, 14, 2, 21) and double transpositions (7, 16, 23) each has as its non ...
A submatrix containing all 24 elements. Cayley table of the symmetric group S 4 showing all 24 permutations of 4 elements ... Symmetric group S4; Permutation notation;
This group has the same rotation axes as T, but with six mirror planes, each through two 3-fold axes. The 2-fold axes are now S 4 (4) axes. T d and O are isomorphic as abstract groups: they both correspond to S 4, the symmetric group on 4 objects. T d is the union of T and the set obtained by combining each element of O \ T with inversion.
Every symmetric group has a one-dimensional representation called the trivial representation, where every element acts as the one by one identity matrix. For n ≥ 2 , there is another irreducible representation of degree 1, called the sign representation or alternating character , which takes a permutation to the one by one matrix with entry ...
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The A000638 (4) = 11 types of subgroups of the symmetric groups S 4 (Subgroups with the same colored cycle graph are bundled together.) Edge colors indicate the quotient of the connected groups' orders: red 2, green 3, blue 4
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