Some sporadic translation planes of order $11^2$

Abstract

In \cite{PK}, the authors constructed a translation plane $\Pi$ of order $11^2$ arising from replacement of a sporadic chain $F'$ of reguli in a regular spread $F$ of $PG(3,11)$. They also showed that two more non isomorphic translation planes, called  $\Pi_1$ and $\Pi_{13}$, arise respectively by derivation and double derivation in $F\setminus F'$ which correspond to a further replacement of a regulus with its opposite regulus and a pair of reguli with their opposite reguli, respectively.  In \cite{AL}, the authors proved that the translation complement of $\Pi$ contains a subgroup isomorphic to $\SL(2,5)$. Here, the full collineation group of each of the planes $\Pi$, $\Pi_1$ and $\Pi_{13}$ is determined.