= Symplectic point stabilizer
{title2=$P_1=U\rtimes(\mathbb F_q^\times\times Sp_{2m-2}(q))$}
For a line $\langle e\rangle$, 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 $(b,t)(b',t')=(b+b',t+t'+B(b,b'))$, giving order $q^{2m-1}$. Its commutator is central with parameter $2B(b,b')$. It is Heisenberg-type in odd characteristic and abelian in characteristic two. The full stabilizer also scales $e,f$ inversely and acts symplectically on $W$.
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