How to find all the homomorphism between two groups?

Question

How many homomorphism $\varphi: S_3\longrightarrow \mathbb{C}^{\times}$ are there?

I don't know how to start, but my intuition tell me that are less than or equal 6.

Answer 1

Hint: $S_3=\langle (13),(23)\rangle$, and each of these generators has order $2$. Given a homomorphism $h:S_3\to\mathbb{C}^{\times}$, $|h(13)|\leq 2$ and $|h(23)|\leq 2$. So, how many elements of order $2$ and order $1$ does $\mathbb{C}^{\times}$ contain?

Answer 2

HINT: Suppose $\varphi: S_3\rightarrow \mathbb{C}^\times$ is a homomorphism. What can we say about the range of $\varphi$? For example, can 7 be in the range of $\varphi$? Think about the orders of the elements of $S_3$ . . .