State and prove Schur's Lemma. Deduce that the centre of a finite group $G$ with a faithful irreducible complex representation $\rho$ is cyclic and that $Z(\rho(G))$ consists of scalar transformations.

Let $G$ be the subgroup of order 18 of the symmetric group $S_{6}$ given by

$$G=\langle(123),(456),(23)(56)\rangle$$

Show that $G$ has a normal subgroup of order 9 and four normal subgroups of order 3 . By considering quotients, show that $G$ has two representations of dimension 1 and four inequivalent irreducible representations of degree 2 . Deduce that $G$ has no faithful irreducible complex representations.

Show finally that if $G$ is a finite group with trivial centre and $H$ is a subgroup of $G$ with non-trivial centre, then any faithful representation of $G$ is reducible on restriction to $H$.