  We study low temperature properties of an atomic spinless interacting Fermi
gas in the trimerized Kagom\'e lattice for the case of two fermions per trimer.
The system is described by a quantum spin 1/2 model on the triangular lattice
with couplings depending on bonds directions. Using exact diagonalizations we
show that the system exhibits non-standard properties of a {\it quantum
spin-liquid crystal}, combining a planar antiferromagnetic order with an
exceptionally large number of low energy excitations.
