The finite groups with an irreducible faithful complex representation

Question

All the groups below are supposed finite, and their representations, complex.

An abelian group admits an irreducible faithful representation iff it is cyclic.
A group has all its non-trivial irreducible representations faithful iff it is simple.

Question: What's the classification of the groups admitting an irreducible faithful representation?

Answer 1

This question admits an answer in the following mathoverflow post:
Which finite groups have faithful complex irreducible representations?