Do you mean a multiplicative character (linear character) or a character of a representation? If you are referring the latter, it is not always true that x(1)=1.
For example, _8 \rightarrow GL_2(\mathbb{Re})" alt="\phi_8 \rightarrow GL_2(\mathbb{Re})" /> can be a matrix representation (verify this), which maps the identity of to identity matrix in , i.e., , where is the character of .