How to compute isometry group directly

Question

Let $M=\Bbb S^1 \times \Bbb R$ be the cylinder. I want to compute its isometry group directly. I know some (possibly all) of its elements:

1. reflection around line perpendicular to $\Bbb R$. by applying this reflection twice it satisfies $R^2=\rm id$.
2. rotations around central line and I think it is $\Bbb S^1$.
3. translations along central line and I think it is $\Bbb R$.

What is the group generated by these elements? Is it $\Bbb S^1\times \Bbb R\times \{I,-I\}$?

Next case, consider $\Bbb S^1\times\Bbb S^1$. For this one we have:

1. Rotations around central vertical line and I think it is $\Bbb S^1$.
2. Rotations inward!! (maybe this one is also $\Bbb S^1$).
3. horizontal plane Reflections: $A^2=id$.
4. vertical plane Reflections: $B^2=id$.

So $\mathrm{Isom}(\Bbb S^1\times\Bbb S^1)=\left<A, B, \Bbb S^1_1, \Bbb S^1_2 \ \big\vert\ A^2=B^2=id\right>=?$.