# non-monomial group with a strong property

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

• $A_5$ 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.
– Dune
Dec 25, 2016 at 9:00
• 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. Dec 25, 2016 at 9:14
• @Dune: Would you mind fleshing that out to an answer? It may be the smallest example? Dec 25, 2016 at 9:16
• @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.
– Dune
Dec 25, 2016 at 9:26
• 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. Dec 29, 2016 at 13:19

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.