Symplectic point stabilizer ($P_1=U⋊({\mathbb{F}}_q^{\times }\times S{p}_{2m-2}(q))$)

For a line $⟨e⟩$, complete $(e,f)$ to a symplectic basis with complementary symplectic space $W$. The unipotent radical has elements $u(b,t)$ sending $e$ to $e$, $w$ to $w-B(w,b)e$ and $f$ to $f+b+te$. Its multiplication is $(b,t)(b^{\prime },t^{\prime })=(b+b^{\prime },t+t^{\prime }+B(b,b^{\prime }))$, giving order $q^{2m-1}$. Its commutator is central with parameter $2B(b,b^{\prime })$. It is Heisenberg-type in odd characteristic and abelian in characteristic two. The full stabilizer also scales $e,f$ inversely and acts symplectically on $W$.