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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?

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

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