Linearly primitive group

Symbol-free definition
A finite group is said to be linearly primitive if it has a faithful irreducible representation over an algebraically closed field of characteristic zero (or equivalently, over the algebraic closure of rationals or over the complex numbers).

Definition with symbols
A group $$G$$ is said to be linearly primitive if there is a homomorphism $$\sigma:G \to GL(V)$$ for some vector space $$V$$ over the complex numbers, such that $$V$$ has no proper nonzero $$G$$-invariant subspace.

Stronger properties

 * Finite simple group

Weaker properties

 * Cyclic-center group: