Can somebody give me concrete problems in finite group representation theory that can be solved using induced representations?
Yes, I understand how induced representations are defined, and I know how to compute their characters, and I know the reciprocity formula. But it bothers me that I have never once needed to used this concept to solve a concrete problem!
For example, maybe somebody can name a finite group whose character table is difficult to construct - until you start looking at representations induced from representations of its subgroups? (Note that these induced representations don't necessarily need to be irreducible to be useful - they can be direct sums of easy-to-find irreps and a hard-to-find irrep. See Jim's answer here.)
If you know any such examples, please let me know, and please don't reveal the full solutions because I would like to try to solve them myself!