  The methyl groups in hexamethylbenzene C$_6$(CH$_3$)$_6$ become magnetically
ordered at the molecular level below 118 K. This is also near the temperature
at which the system structurally transitions from triclinic to a unique
near-cubic phase. High-precision measurements of the near-static dielectric
constant reveal that the structural phase transition is actually comprised of
four successive transformations upon cooling at T$_1$=110.7 K, T$_2$=109.5 K,
T$_3$=109.1 K and T$_4$=107.8 K. In contrast to warming, only two transitions
occur at T'$_4$=119.2 K and T'$_1$=120.9 K. The methyl groups in the near-cubic
phase become slightly distorted according to existing neutron powder
diffraction measurements. Analysis of the 2$^6$=64 possible spin orientation
configurations of the methyl groups reveal a 20-fold ground state degeneracy
presiding in each molecule rendering the system to become highly unstable. From
such, it is interpreted that T$_1$ and T$_2$ are temperatures at which the
molecules successively lower their symmetry to remove the energy degeneracy
which involve methyl group elongation and further out-of-plane tilting. This
triggers the system to phase transition into the near-cubic phase which
involves a shearing of the molecular planes and partial rotation of the methyl
groups at T$_3$ and T$_4$. We interpret the low temperature near-cubic phase to
be a manifestation of Jahn-Teller distortions based on energy degeneracies of
the orbital motion of protons in each molecule and suggest that the metastable
nature of the phase transition originates from the methyl groups requiring a
larger amount of energy to order than to disorder. Our findings help explain
why many unusual structural phase transitions occur at low temperatures in
other molecular crystals possessing periodic motion of protons.
