A Classification of Autoparatopisms of Latin Cubes

A paratopism is an action on a Latin hypercube of dimension d and order n which is an element of the wreath product $S_n\wr S_{d+1}$. A paratopism is said to be an autoparatopism if there is at least one Latin hypercube which is mapped to itself under the action of the paratopism. In this paper we classify autoparatopisms of Latin cubes given d = 3 and $n\in {\mathbb{Z}}^{+}$, upto the conjugacy in $S_n\wr S_4$. In order to achieve this objective, we prove that given an autoparatopism $\sigma \in S_n\wr S_{d+1}$, all the conjugates of $\sigma$ are autoparatpisms. Also, an important condition is presented, which states that the cycle structure of the $\delta$ in the element ${\sigma}= ({\alpha}_1,{\alpha}_2,{\alpha}_3,{\alpha}_4;{\delta})$of$S_n \wr S_4$ determines the conjugacy.