A tropical analogue of the Hessian group

We investigate a tropical analogue of the Hessian group $G_{216}$, the group of linear automorphisms acting on the Hesse pencil. Through the procedure of ultradiscretization, the group law on the Hesse pencil reduces to that on the tropical Hesse pencil. We then show that the dihedral group $\mathcal{D}_3$ of degree three is the group of linear automorphisms acting on the tropical Hesse pencil.