The groups are finite and representations are over $\mathbb{C}$.
A group $G$ is said to be monomial if any irreducible representation of $G$ of dimension $>1$ is induced from one-dimensional representation of some subgroup.
Of course, there are non-monomial groups. But among them, I want to consider examples of groups with further property:
Q. Are there non-monomial groups, whose any irreducible representation of dimension $>1$ is not induced from irreducible representation of any subgroup?
Writing this symbolically, is there a finite group $G$ with $\chi\in$ Irr$(G)$ such that
(1) $\chi(1)>1$.
(2) There is no pair $(H,\psi)$ with $H$ proper subgroup of $G$, $\psi\in $Irr$(H)$ and $\chi=\mbox{Ind}_H^G \psi$