Let $G$ be a discrete group and $\beta (G)$ denote the Stone-Cech compactification of $G$, a right topological semigroup. By a multiplicative character, I mean a mapping that preserves multiplication but it is not necessarily continuous.

1- Is there any example of a multiplicative character $\gamma :\beta (G)\to \mathbb{T}$ that is not continuos?

2- Is there any concrete example of a multiplicative character $\gamma :\beta (G)\to \mathbb{T}$? I know how to build abstract examples but don't know of any concrete example.