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# Encyclopaedia of DesignTheory: A character table

Let G be the symmetric group of degree 3, the group of all permutations of the set {1,2,3}.

There are three conjugacy classes in G. The class C1 consists of the identity permutation, denoted (1). The class C2 consists of the two cyclic permutations (1,2,3) and (1,3,2). The class C3 consists of the three transpositions (1,2), (1,3) and (2,3).

The number of irreducible representations is the same, namely 3. They are as follows:

• The trivial representation R1 which maps every element to the 1×1 matrix (1).
• The "sign representation" R2 which maps even permutations (elements of C1 and C2) to (+1) and odd permutations (elements of C3) to (-1).
• A two-dimensional representation R3 on the set of all vectors (x,y,z) with x+y+z = 0, where elements of G act by permuting the coordinates. The character of this representation is obtained by subtracting one from the number of fixed points of each element of G.

Thus the character table of G is as follows, where Xi is the character of the representation Ri:

 C1 C2 C3 X1 +1 +1 +1 X2 +1 +1 -1 X3 +2 -1 0

Peter J. Cameron
6 August 2002