  In the present work, we report a detailed description of the symmetry
properties of the eight-atomic molecule ethane, with the aim of facilitating
the variational calculations of rotation-vibration spectra of ethane and
related molecules. Ethane consists of two methyl groups $\text{CH}_3$ where the
internal rotation (torsion) of one $\text{CH}_3$ group relative to the other is
of large amplitude and involves tunneling between multiple minima of the
potential energy function. The molecular symmetry group of ethane is the
36-element group $G_{36}$ but the construction of symmetrized basis functions
is most conveniently done in terms of the 72-element extended molecular
symmetry group $G_{36}\text{(EM)}$. This group can subsequently be used in the
construction of block-diagonal matrix representations of the ro-vibrational
Hamiltonian for ethane. The derived transformation matrices associated with
$G_{36}\text{(EM)}$ have been implemented in the variational nuclear motion
program TROVE (Theoretical ROVibrational Energies). TROVE variational
calculations will be used as a practical example of a $G_{36}\text{(EM)}$
symmetry adaptation for large systems with a non-rigid, torsional degree of
freedom. We present the derivation of irreducible transformation matrices for
all 36 (72) operations of $G_{36}\text{(M)}$ ($G_{36}\text{(EM)}$) and also
describe algorithms for a numerical construction of these matrices based on a
set of four (five) generators. The methodology presented is illustrated on the
construction of the symmetry-adapted representations both of the potential
energy function of ethane and of the rotation, torsion and vibration basis set
functions.
