2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/101325We consider the classification problem for compact Lie groups $G\subset U(n)$ which are generated by a single conjugacy class with a fixed number $N$ of distinct eigenvalues. We give an explicit classification when N=3, and apply this to extract information about Galois representations and braid group representations.30 pages. version 3: many typos fixed, section 6 substantially reorganized. To appear in Int. Math. Res. NotRepresentation TheoryGeometric Topology22E15; 22F36; 20F36text