  We find identities between theta constants with rational characteristics
evaluated at period matrix of $R,$ a cyclic 3 sheeted cover of the sphere with
$3k$ branch points $\lambda_1...\lambda_{3k}.$ These identities follow from
Thomae formula \cite{BR}. This formula expresses powers of theta constants as
polynomials in $\lambda_1...\lambda_{3k}.$ We apply the representation of the
symmetric group to find relations between the polynomials and hence between the
associated theta constants.
