A group character of a finite-dimensional group representation is its matrix trace, regarded
as a function on the group.
For a representation 
,
it is 
.
Irreducible group characters are those belonging to irreducible
representations. The group orthogonality
theorem gives their inner-product relations.
All members of the same conjugacy class in the same representation have the same character. Members of other conjugacy
classes may also have the same character, however. An (abstract) group
can be identified by a listing of the characters of its various representations,
known as a character table. However, there exist
nonisomorphic groups which nevertheless have the same character table, for example 
(the symmetry group of
the square) and 
(the quaternion group).
