Normalizer of a split Cartan subgroup (awaiting review)

The normalizer of a split Cartan subgroup of $\GL_2(\F_p)$ is a maximal subgroup of $\GL_2(\F_p)$ that contains a split Cartan subgroup with index 2. For $p>2$ such a group is in fact the normalizer in $\GL_2(\F_p)$ of the split Cartan subgroup it contains, but for $p=2$ this is not the case (the split Cartan subgroup of $\GL_2(\F_2)$ is already normal).

The label Ns identifies the subgroup generated by the split Cartan subgroup Cs of diagonal matrices and the matrix \[ \begin{pmatrix}0&1\\1&0\end{pmatrix}. \] Every normalizer of a split Cartan subgroup is conjugate to the group Ns.

The label Ns.a.b identifies the proper subgroup of Ns generated by the matrices \[ \begin{pmatrix}a&0\\0&1/a\end{pmatrix}, \begin{pmatrix}0&b\\-r/b&0\end{pmatrix}, \] where $a$ and $b$ are minimally chosen positive integers and $r$ is the least positive integer generating $(\Z/p\Z)^\times\simeq \F_p^\times$.

The label Ns.a.b.c identifies the proper subgroup of the normalizer of the split Cartan subgroup generated by the matrices \[ \begin{pmatrix}a&0\\0&1/a\end{pmatrix}, \begin{pmatrix}0&b\\-1/b&0\end{pmatrix}, \begin{pmatrix}0&c\\-r/c&0\end{pmatrix} \] where $a$ and $b$ are minimally chosen positive integers and $r$ is the least positive integer generating $(\Z/p\Z)^\times\simeq \F_p^\times$, as defined in [arXiv:1504.07618, 10.1017/fms.2015.33, MR:3482279]}.