Cubic maps from the group of order $3$

The purpose of this note is to classify unital cubic maps from the cyclic group of order $3$ into an arbitrary non-abelian group. We show that the universal group admitting a unital cubic map from the cyclic group of order $3$ is infinite, give a concrete presentation and provide an infinite representation of it in ${\mathrm{P}\mathrm{S}\mathrm{L}}_3(\mathbb{C})$, whose image is an arithmetic lattice commensurable with ${\mathrm{P}\mathrm{S}\mathrm{L}}_3(\mathbb{Z}[\omega ])$, where $\omega$ is a primitive cube root of unity. As a consequence we obtain the existence of finite nilpotent groups of arbitrarily large nilpotency class admitting a unital cubic map from $C_3$ whose image generates the group.