  We introduce the notion of Mixed Symmetry Quantum Phase Transition (MSQPT) as
singularities in the transformation of the lowest-energy state properties of a
system of identical particles inside each permutation symmetry sector $\mu$,
when some Hamiltonian control parameters $\lambda$ are varied. We use a
three-level Lipkin-Meshkov-Glick (LMG) model, with $U(3)$ dynamical symmetry,
to exemplify our construction. After reviewing the construction of $U(3)$
unirreps using Young tableaux and Gelfand basis, we firstly study the case of a
finite number $N$ of three-level atoms, showing that some precursors
(fidelity-susceptibility, level population, etc.) of MSQPTs appear in all
permutation symmetry sectors. Using coherent (quasi-classical) states of $U(3)$
as variational states, we compute the lowest-energy density for each sector
$\mu$ in the thermodynamic $N\to\infty$ limit. Extending the control parameter
space by $\mu$, the phase diagram exhibits four distinct quantum phases in the
$\lambda$-$\mu$ plane that coexist at a quadruple point. The ground state of
the whole system belongs to the fully symmetric sector $\mu=1$ and shows a
four-fold degeneracy, due to the spontaneous breakdown of the parity symmetry
of the Hamiltonian. The restoration of this discrete symmetry leads to the
formation of four-component Schr\"odinger cat states.
