Isometries of Sine Curve

Question

I am trying to find all of the isometries of a sine curve. I found four isometries (the identity, rotating by 180 degrees, shifting by a multiple of $2 \pi,$ and reflecting over a vertical line). My question is - how do I create a multiplication table for these operations? I tried looking at how different transformations affect points (like a translation followed by a rotation), but haven't been able to come up with a solution.