Let G be a finite group. A class function of G is a function from G into the complex numbers (or a subfield of the complex numbers) that is constant on conjugacy classes of G. Addition, multiplication, and scalar multiplication of class functions are defined pointwise. Thus the set of all class functions of G is an algebra (or ring, or vector space).
Class functions and (virtual) group characters
Every mapping with source G that is constant on conjugacy classes of G is called a class function of G. Differences of characters of G are called virtual characters of G.
Class functions occur in a natural way when one deals with characters. For example, the central character of a group character is only a class function.
Every character is a virtual character, and every virtual character is a
class function.
Any function or operator that is applicable to a class function can of
course be applied to a (virtual) group character. There are functions
only for (virtual) group characters, like IsIrreducible, which doesn't
make sense for a general class function, and there are also functions
that do not make sense for virtual characters but only for characters,
like Determinant.
Class functions as mappings
In GAP, class functions of a group G are mappings (see chapter
Mappings) with source G and range Cyclotomics (or a subfield).
All operators and functions for mappings (like Image Image, PreImages
PreImages) can be applied to class functions.
Note, however, that the operators * and ^ allow also other
arguments than mappings do (see Operators for Class Functions).
GAP 3.4.4