Consequences of $G$ having exactly $3$ irreducible characters

Question

Suppose a finite group $G$ has exactly irreducible characters $\chi_1 = \mathbb{I}, \chi_2,\chi_3$.

i) Show that G is soluble and deduce it has a non-trivial one-dimensional character $\chi_2$.

ii) Show that $\dim \chi_3 \in \{1,2\}$.

I cannot get anywhere with either part so hints only please.

Answer 1

Can you find all finite groups with exactly $3$ conjugacy classes? Hint: there are only two of them.