= Symplectic Lagrangian stabilizer
{title2=$P_m=\operatorname{Sym}_m(q)\rtimes GL_m(q)$}
With a <Lagrangian subspace> first in a paired <symplectic basis>, its stabilizer consists of $\begin{pmatrix}A&AS\\0&A^{-\mathsf T}\end{pmatrix}$ with $S$ symmetric. The additive symmetric-matrix <unipotent radical> has order $q^{m(m+1)/2}$; the <Levi subgroup> is $GL_m(q)$, acting by $S\mapsto ASA^{\mathsf T}$. Its index in $Sp_{2m}(q)$ is $\prod_{i=1}^m(q^i+1)$, the number of Lagrangian subspaces.
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