Irreducible representation which is not induced one Is there any example of a finite group $G$ with an irreducible representation $\rho$ which is not induced from an irreducible representation of any (proper) subgroup?
 A: The trivial representation (or any other $1$-dimensional representation) can never be induced from a representation of a proper subgroup, since when you induce a representation its dimension gets multiplied by the index of your subgroup.
A: The group $G=S_5$ has six Sylow $5$-subgroups, each with a normalizer of order twenty. One such normalizer is $N=\langle(12345),(2354)\rangle$. It is easy to check that the action of $G$ on $G/N$ is doubly transitive. Therefore $\operatorname{Ind}_N^G 1$ is the direct sum of the trivial character of $G$ and a $5$-dimensional irreducible character $\chi$. 
The character $\chi$ cannot be constructed by inducing an irreducible character of any subgroup of $S_5$. Because $5$ is a prime the only possibility would be to induce it from a 1-dimensional character of an index five subgroup $H$. Such a subgroup would have to be $S_4$. The group $S_4$ has the two obvious 1-dimensional characters, the trivial character and the sign character. But inducing either of those to $S_5$ gives a direct sums of the corresponding 1-dimensional character of $S_5$ and an irreducible $4$-dimensional character,
