(a) Let $G$ be a finite group and let $\rho: G \rightarrow \mathrm{GL}_{2}(\mathbb{C})$ be a representation of $G$. Suppose that there are elements $g, h$ in $G$ such that the matrices $\rho(g)$ and $\rho(h)$ do not commute. Use Maschke's theorem to prove that $\rho$ is irreducible.

(b) Let $n$ be a positive integer. You are given that the dicyclic group

$$G_{4 n}=\left\langle a, b: a^{2 n}=1, a^{n}=b^{2}, b^{-1} a b=a^{-1}\right\rangle$$

has order $4 n$.

(i) Show that if $\epsilon$ is any $(2 n)$ th root of unity in $\mathbb{C}$, then there is a representation of $G_{4 n}$ over $\mathbb{C}$ which sends

$$a \mapsto\left(\begin{array}{cc} \epsilon & 0 \\ 0 & \epsilon^{-1} \end{array}\right), \quad b \mapsto\left(\begin{array}{cc} 0 & 1 \\ \epsilon^{n} & 0 \end{array}\right)$$

(ii) Find all the irreducible representations of $G_{4 n}$.

(iii) Find the character table of $G_{4 n}$.

[Hint: You may find it helpful to consider the cases $n$ odd and $n$ even separately.]