3 added 396 characters in body

If we know the character of a representation (of a finite group) over C (field of complex numbers), is it possible to recover the representation itself?

This is clearly possible if we know all the irreducible representations of the group. But what if we don't know them?

ADDED: 1) We know the group. By this I mean we have the "multiplication table" of the group. 2) We don't know the irreducible representations. We are only given a character of a representation. 3) We want to obtain a concrete realization of a representation yielding the given character. By this I mean a matrix for each element of the group. 4) Finally, we don't care about efficiency.