Given a finite group and a normal subgroup, does there always exist an irreducible complex representation, whose kernel is this normal subgroup?


Sorry, just it was just mentioned that this is a duplicate. See [Which finite groups have faithful complex irreducible representations?](https://mathoverflow.net/questions/57129/which-finite-groups-have-faithful-complex-irreducible-representations).



Does every normal subgroup appear as a kernel of an irreducible representation? 
take the canonical quotient $q:G\rightarrow G/H$ and $\lambda :G/H\rightarrow U(n)$ to be the left-regular representation of $G/H$, where $n=|G/H|$, then the composition is the map that you are looking for, unless I am missing something...


**Edit:** if you consider irreducible representations, then the answer is no for the group $\mathbb{Z}\_2\oplus\mathbb{Z}\_2\oplus\mathbb{Z}\_2$.


