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:

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

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Peter J. Cameron
6 August 2002