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In this work we compute the Dixmier invariants and bitangents of the planequartics with 3,6 or 9-cyclic automorphisms, we find that a quartic curve with6-cyclic automorphism will have 3 horizontal bitangents which form an asysgetictriple. We also discuss the linear matrix representation problem of suchcurves, and find a degree 6 equation of 1 variable which solves the symbolicsolution of the linear matrix representation problem for the curve with6-cyclic automorphism.

Invariants, Bitangents and Matrix Representations of Plane Quartics with 3-Cyclic Automorphisms