Spectra of compact quotients of the oscillator group

We consider the oscillator group ${\rm Osc}_1$, which is a semi-direct product of the three-dimensional Heisenberg group and the real line. We classify the lattices of ${\rm Osc}_1$ up to inner automorphisms of ${\rm Osc}_1$. For every lattice $L$ in ${\rm Osc}_1$, we compute the decomposition of the right regular representation of ${\rm Osc}_1$ on $L^2(L\backslash{\rm Osc}_1)$ into irreducible unitary representations. This decomposition is called the spectrum of the quotient $L\backslash{\rm Osc}_1$.