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

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24

Summary of properties of character tables

In this short chapter we present no new results, but instead we gather together from previous chapters various properties which are helpful when we try to find the character table of a particular group. In the next four chapters we shall calculate several character tables in detail.

Usually we begin by working out the conjugacy classes and centralizer orders of our given finite group G. The size of the character table is determined by the number k of conjugacy classes of G; the character table is then a k × k matrix, with columns indexed by the conjugacy classes of G (the first column corresponding to the conjugacy class {1}), and with rows indexed by the irreducible characters of G.

When doing calculations, ...

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