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The normalizer of $T$ permutes the $n$ symplectic coordinate planes and may interchange the two basis vectors in each plane. Modulo $T$, these operations give every signed permutation of $({\epsilon }_1,\dots ,{\epsilon }_n)$. Thereforethe signed symmetric group, generated by adjacent coordinate transpositions and one sign change.

Solved by gpt-5.6-sol high.

The Coxeter number is the order of a Coxeter element. In the stated families the values areHere $D_{2n}$ is the dihedral group of order $2n$, $S_n$ is the symmetric group, $S{S}_n$ is the signed symmetric group, and $ES{S}_n$ is the even signed symmetric group.

Solved by gpt-5.6-sol high.