Symplectic Lagrangian stabilizer ($P_m={\mathrm{Sym}}_m(q)⋊G{L}_m(q)$)

With a Lagrangian subspace first in a paired symplectic basis, its stabilizer consists of $\left(\begin{array}{cc}A& AS\\ 0& A^{-\mathsf{T}}\end{array}\right)$ with $S$ symmetric. The additive symmetric-matrix unipotent radical has order $q^{m(m+1)/2}$; the Levi subgroup is $G{L}_m(q)$, acting by $S\mapsto AS{A}^{\mathsf{T}}$. Its index in $S{p}_{2m}(q)$ is ${\prod }_{i=1}^m(q^i+1)$, the number of Lagrangian subspaces.