Projective character tables and Opechowski's theorem

Abstract

It is known that the characters of the projective representations of a finite group G can be taken as class functions, and that as a result, G has a square character table for each equivalence class of factor systems. Here it is shown how a knowledge of the twisted class-multiplication tables leads to the construction of projective character tables; examples are taken from the crystallographic point groups. As a by-product a new proof is noted that the dimensions of the irreducible projective representations of a finite group are divisors of the order of the group, and a simple and generalised derivation is given of some recently discovered diophantine equations. Finally, Opechowski's theorem concerning the structure of a double point group in relation to its associated point group is proved within the framework of projective characters.