What is the group generated by Hadamard gate and T-gate?
Or, in terms of Algebra:
Let $U$ be the multiplicative group of 2-by-2 unitary matrices, and let $P$ be $\{e^{i\phi}I : \phi \in [0,2\pi)\}$. Let $G = U/P$. Then $G$ is isomorphic to $SO(3)$.
Let $H = \frac{1}{\sqrt2}\begin{bmatrix}1 & 1 \\ 1 & -1\end{bmatrix}$, $S = \begin{bmatrix}1 & 0 \\ 0 & i\end{bmatrix}$, and $T = \begin{bmatrix}1 & 0 \\ 0 & e^{i\pi/4}\end{bmatrix}$ be members of $G$.
The subgroup of $G$ generated by $H$ and $S$ is the chiral octahedral group $O$, or in terms of abstract algebra, the symmetric group $S_4$. $S$ corresponds to cycle $(0 \space 1 \space 2 \space 3)$, and $H$ corresponds to cycle $(0 \space 1)$.
But what is the subgroup $A$ of $G$ generated by $H$ and $T$? It is not isomorphic to a finite subgroup of $SO(3)$, so it must be countably infinite.
Specifically, my questions are:
Is $A$ dense in $SO(3)$?
For every prime number $p$, does there exist a member $R$ of $A$ such that $R^p = \begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix}$?
What exactly are the members of $A$? Is there a geometric intuition to depict $A$?