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Aug 27, 2023 at 22:28 comment added Gerry Myerson @Stanley, for, say, $n=3$, isn't it known what finite groups have exactly three conjugacy classes (and, thus, exactly three irreducible degrees)? cefns.nau.edu/~falk/old_classes/511/extras/threeclass.html says the only ones are the group of order three, and the symmetric group on three letters. See also math.stackexchange.com/questions/1581996/…
Jun 17, 2023 at 0:40 vote accept semisimpleton
Jun 16, 2023 at 22:39 comment added Stanley Yao Xiao I think the general question of finding all positive integer solutions to the system of congruences might be beyond current technology, but the restricted question of looking for solutions corresponding to irreducible degrees of a finite group might be tractable. Thus I believe this is an interesting question, because it seems to lie on the boundary of what’s currently approachable
Jun 16, 2023 at 17:41 history answered Peter Mueller CC BY-SA 4.0