1)Let $G$ be a group of size $48$ with centre consisting of the identity element only. Show that $G$ has a conjugacy class of size $3$ .
My attempt: I know $C(g)$ centraliser of $g$ is a subgroup of $G$, so its size must divide that of $G$. And that $|cl(g)|=|G|/|c(g)|$ , but how am I sure there exists a $g\in G$ such that$|c(g)|=16$?
2)Also say I had a group of size 48 and at least one conjugacy class of size 3 does that imply the centre only consists on the identity element?
Any help on 2) would be appreciated thanks.