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Representations and Characters of Groups, Second Edition by Martin Liebeck, Gordon James

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18

Some elementary character tables

We now illustrate the techniques we have presented so far by constructing the character tables of several groups, including the groups S4 and A4, and all dihedral groups.

18.1 The group S4

In Example 17.4, we produced three irreducible characters χ1, χ2, χ3 of S4 by lifting characters of the factor group S4/V4. We shall now use Proposition 17.14, which deals with the product of a character with a linear character, to complete the character table of S4.

Let χ4 be the character

image

which is given in Proposition 13.24. By Proposition 17.14, the product χ4χ2 is also a character of S4. The values of χ2, χ4 and χ4χ2

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