Real Heisenberg quotient has no faithful finite-dimensional representation
= Real Heisenberg quotient has no faithful finite-dimensional representation
Let $G$ be the real upper unitriangular three-by-three group, let $Z$ be its centre, and let $Z_0$ be the integer subgroup of $Z$. Every finite-dimensional continuous complex representation of $G/Z_0$ kills the central circle $Z/Z_0$, and hence is not faithful.