Abstract
This chapter defines the representations of finite groups and considers successively the cases of crystallographic point groups in one, two and three dimensions and of space groups. It introduces the notions of irreducible representations. The character tables and irreducible representations are given for the 32 crystallographic point groups in three dimensions. The transformation properties of tensors are also considered. In the last section, the accompanying software Tenχar is introduced. This software can be used to determine the irreducible representations of finite point groups in three dimensions (the 32 crystallographic groups and the groups of the quasicrystalline phases) and the independent components of a tensor of any rank for each of these groups.
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