(a) Prove that if there exists a faithful irreducible complex representation of a finite group $G$, then the centre $Z(G)$ is cyclic.

(b) Define the permutations $a, b, c \in S_{6}$ by

$$a=\left(\begin{array}{lll} 1 & 2 & 3 \end{array}\right), b=\left(\begin{array}{lll} 4 & 5 & 6 \end{array}\right), c=\left(\begin{array}{ll} 2 & 3 \end{array}\right)(45),$$

and let $E=\langle a, b, c\rangle$.

(i) Using the relations $a^{3}=b^{3}=c^{2}=1, a b=b a, c^{-1} a c=a^{-1}$ and $c^{-1} b c=b^{-1}$, prove that $E$ has order 18 .

(ii) Suppose that $\varepsilon$ and $\eta$ are complex cube roots of unity. Prove that there is a (matrix) representation $\rho$ of $E$ over $\mathbb{C}$ such that

$$a \mapsto\left(\begin{array}{cc} \varepsilon & 0 \\ 0 & \varepsilon^{-1} \end{array}\right), \quad b \mapsto\left(\begin{array}{cc} \eta & 0 \\ 0 & \eta^{-1} \end{array}\right), \quad c \mapsto\left(\begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right)$$

(iii) For which values of $\varepsilon, \eta$ is $\rho$ faithful? For which values of $\varepsilon, \eta$ is $\rho$ irreducible?

(c) Note that $\langle a, b\rangle$ is a normal subgroup of $E$ which is isomorphic to $C_{3} \times C_{3}$. By inducing linear characters of this subgroup, or otherwise, obtain the character table of $E$.

Deduce that $E$ has the property that $Z(E)$ is cyclic but $E$ has no faithful irreducible representation over $\mathbb{C}$.