  We investigate the invariants of the $25$-dimensional real representation of
the group ${\bf SO}(3)\wr{\bf Z}_2$ given by the left and right actions of
${\bf SO}(3)$ on $5\times 5$ matrices together with matrix transposition; the
action on column vectors is the irreducible $5$-dimensional representation of
${\bf SO}(3)$. The $25$-dimensional representation arises naturally in the
study of nematic liquid crystals, where the second-rank orientational order
parameters of a molecule are represented by a symmetric $3\times3$ traceless
symmetric matrix, and where a rigid rotation in ${\bf R}^3$ induces a linear
transformation of this space of matrices. The entropy contribution to a free
energy density function in this context turns out to have ${\bf SO}(3)\wr{\bf
Z}_2$ symmetry. Although it is unrealistic to expect to describe the complete
algebraic structure of the ring of invariants, we are able to calculate the
Molien series giving the number of linearly independent invariants at each
homogeneous degree, and to express this as a rational function indicating the
degrees of invariant polynomials that constitute a basis of 19 primary
invariants. The algebra of invariants up to degree 4 is investigated in detail.
