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Jul 5, 2020 at 20:39 vote accept Sebastien Palcoux
Jun 22, 2020 at 17:33 comment added John Shareshian I think that when $n=17$, you can use the same $\chi$ as in the answer (restricted to $A_n)$ and $w$ of shape $(9,4,4)$ to get an example in a simple group. There are likely infinitely many examples like this.
Jun 22, 2020 at 17:26 history edited John Shareshian CC BY-SA 4.0
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Jun 22, 2020 at 17:22 comment added Sebastien Palcoux @JohnShareshian Do you expect the existence of an example with $G$ an alternating group (or more generally a simple group)?
Jun 22, 2020 at 16:25 history answered John Shareshian CC BY-SA 4.0