So this is an old prelim question and I have some questions. Let me first state the pre lim problem. If $G$ is a nonabelian group of order 57 how many 1-dimensional characters does $G$ have.
Now I know that up to isomorphism there is only one such group $G$ which can be realized as a semi direct product of $Z_{19}$ and $Z_3$. My question is regarding the definition of character. I've seen it defined as maps from $G\rightarrow \mathbb{C}^\times$ or generally to maps $G\rightarrow F$ where $F$ is a field, which is the correct definition? If its the first case how do I count such maps. Sorry I'm unfamiliar with representation/character theory, so assume that I'm completely uninformed on this matter. I was thinking of defining the maps into $\mathbb{C}^\times$ on $Z_{19}$ and $Z_3$ separately and combing them somehow, but now sure of my constructions. There's probably some theorems/lemmas from representation theory that would make this easier, but again I'm not familiar with them. Any help would be appreciated.
For counting (complex) linear characters, you don't even have to construct them all. A standard result in character theory, see for instance Isaacs (1994, Corollary 2.23), says that such number equals the index of the commutator subgroup $\lvert G : G' \rvert$. Hence it equals three, as you've already know.