Classification of degree-n Riordan representations of S3 over GF(3^q)

Classify all degree-n Riordan representations of the symmetric group S3 over the Galois field GF(3^q) for integers q ≥ 2 and n ≥ 2, where a degree-n Riordan representation arises by composing a Riordan array representation over GF(3^q) with the truncation map R(GF(3^q)) → Rn(GF(3^q)).

Background

Corollary 8 establishes the existence of faithful Riordan representations of S3 over GF(3^q) for all q, generalizing the Z3 case. The authors successfully handle degree-2 over Z3, but a broader classification over GF(3^q) and higher degrees remains unresolved.

This question seeks to describe all nonequivalent degree-n Riordan representations over varying finite fields GF(3^q), extending the structural analysis given in Section 3.