  In this work we compute the Dixmier invariants and bitangents of the plane
quartics with 3,6 or 9-cyclic automorphisms, we find that a quartic curve with
6-cyclic automorphism will have 3 horizontal bitangents which form an asysgetic
triple. We also discuss the linear matrix representation problem of such
curves, and find a degree 6 equation of 1 variable which solves the symbolic
solution of the linear matrix representation problem for the curve with
6-cyclic automorphism.
