Let $G$ be a finite group with order 168 such that for each element $g \in G$ wih order $7$, the centralizer $C_G (g)$ is equal to $\langle g \rangle$. Compute the number of conjugacy classes of elements $g$ of order $7$ and find the size of the conjugacy class for such an element.
I tried to begin with the class equation, but didn't have any luck. I don't know if I will need to use the definition of a Sylow subgroup at some point here.
I would appreciate any help.