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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 $\chi=\mbox{Ind}_H^G \psi$

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    $\begingroup$ $A_5$ should be an example: There are irreducible characters of degree 3, but $A_5$ has no proper subgroup of index dividing 3, so they cannot be induced. $\endgroup$
    – Dune
    Dec 25, 2016 at 9:00
  • $\begingroup$ Would the 5-dimensional irreducible reps of $G=S_5$ serve in the role $\chi$? Because $5$ is a prime we need $H$ to have index five, so essentially $H=S_4$, and $\psi$ is 1-dimensional. But IIRC the characters we get from inducing a linear character of $S_4$ to $S_5$ are not irreducible. $\endgroup$
    – Jyrki Lahtonen
    Dec 25, 2016 at 9:14
  • $\begingroup$ @Dune: Would you mind fleshing that out to an answer? It may be the smallest example? $\endgroup$
    – Jyrki Lahtonen
    Dec 25, 2016 at 9:16
  • $\begingroup$ @JyrkiLahtonen: I just guessed the group, and looked up the facts in tables. But if I find some time I will think about smaller examples. $\endgroup$
    – Dune
    Dec 25, 2016 at 9:26
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    $\begingroup$ I haven't found a nonabelian example in which no irreducible representation is induced from a proper subgroup, so it might be an interesting question whether such an example exists. $\endgroup$
    – Derek Holt
    Dec 29, 2016 at 13:19

1 Answer 1

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Looking at the list of small groups we see that all groups up to order 23 are monomial (all but the alternating group $A_4$ are even supersolvable). There is a unique non-monomial group of order 24, namely $\mathrm{SL}(2,3)$. This group has irreducible characters of degree two (see here), but none of them are induced by smaller ones, since there is no subgroup of index two in $\mathrm{SL}(2,3)$. So this is the smallest example of a group you are looking for.

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