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

Theta constants identities for Jacobians of cyclic 3-sheeted covers of the sphere and representations of the symmetric group