= Nonfaithful representations of the infinite dihedral group
Every nonfaithful finite-dimensional complex representation of the infinite dihedral group is a direct sum of subrepresentations of dimension at most two. Its kernel contains a nonzero power of the rotation, so the representation factors through a <dihedral group>[finite dihedral group]. <Maschke's theorem> makes that quotient representation semisimple, and all <irreducible complex representations of a finite dihedral group> have dimension at most two.
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