Let a rank-3 matrix representation of a group $G$ generated by two group elements. Say one is a group as special unitary SU(2), $$ A=\begin{pmatrix} 1 & 0 & 0 \\ 0 & \cos(\frac{\theta}{2})+{i a_z} \sin(\frac{\theta}{2}) & (i a_x -a_y) \sin(\frac{\theta}{2}) \\ 0 & (i a_x +a_y) \sin(\frac{\theta}{2}) & \cos(\frac{\theta}{2})-{i a_z} \sin(\frac{\theta}{2}) \end{pmatrix}, $$ with $\theta$ is 4$\pi$ periodic, and $(a_x,a_y,a_z)$ is a unit vector [on a $S^2$].
The other is a $\mathbb{Z}_2$ element $$ B=\left( \begin{array}{ccc} -1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \\ \end{array} \right)$$ Notice the SU(2) generated by $A$ and the $\mathbb{Z}_2$ generated by $B$ do not commute. But they are all within the special unitary SU(3).
What is this full group $G$ generated by $A$ and $B$, as a subgroup of SU(3)? (All $g \in G$, then $g=A^n B^m A^l B^o ...$.)
Clarification: The main point is not just trying to write down an abstract group notation for this $G$. Roughly speaking, there is at least a $\mathbb{Z}_4$ subgroup in SU(2), such that $\mathbb{Z}_4 \subset$ SU(2), that has nontrivial multiplication with the $\mathbb{Z}_2$, thus this $G$ contains a dihedral group $\mathbb{Z}_4 \rtimes \mathbb{Z}_2$. However, the full group $G$ may not be simply SU(2) $\rtimes \mathbb{Z}_2$(? Up to your proper definition). There can be an additional finite group sector outside this $\mathbb{Z}_2$, so roughly speaking the order of group can be larger than |SU(2)| $\cdot |\mathbb{Z}_2|$. Namely, the order of group can be |SU(2)| $\cdot |K|$, where $K \supset \mathbb{Z}_2$ and $|K| > |\mathbb{Z}_2|$.
What may be the additional finite group sector $K$, if there is any?