I'm trying to find the minimal generating set for a square prism under reflection. This group is $\text{Dih}_4\times Z_2$.
Geometrically, the set $\{a,b,c\}$ where $a$ is a rotation of $90^\circ$ about the axis through the square faces, $b$ is a rotation of $180^\circ$ about an axis through two rectangular faces, and $c$ is a reflection about some plane, would generate the group. After all, $\{a,b\}$ generates $\text{Dih}_4$ and $\{c\}$ generates $Z_2$. However, I don't know how to prove that there does not exist two elements $\{x,y\}$ which generate the whole group.
Is this even the case? In any case, how can it be proven or disproven?