The Fischer-Clifford matrices of a maximal subgroup of $Fi^{\prime}_{24}$

The Fischer group $Fi_{24}^{\prime}$ is the largest sporadic simple Fischer group of order

The group $Fi_{24}^{\prime}$ is the derived subgroup of the Fischer $3$-transposition group $Fi_{24}$ discovered by Bernd Fischer. There are five classes of elements of order 3 in $Fi_{24}^{\prime}$ as represented in ATLAS by $3A$, $3B$, $3C$, $3D$ and $3E$. A subgroup of $Fi_{24}^{\prime}$ of order $3$ is called of type $3X$, where $X \in \{A,B,C,D,E \}$, if it is generated by an element in the class $3X$. There are six classes of maximal 3-local subgroups of $Fi_{24}^{\prime}$ as determined by Wilson. In this paper we determine the Fischer-Clifford matrices and conjugacy classes of one of these maximal 3-local subgroups $\bar{G} := N_{Fi_{24}^{\prime}}(\langle N\rangle ) \cong 3^7{\cdot}O_7(3)$, where $N \cong 3^7$ is the natural orthogonal module for $\bar{G}/N \cong O_7(3)$ with $364$ subgroups of type $3B$ corresponding to the totally isotropic points. The group $\bar{G}$ is a nonsplit extension of $N$ by $G \cong O_7(3)$.