Using the simulation website below, you can see the symmetry elements (axes and planes) of the molecule ethane. http://symmetry.otterbein.edu/gallery/index.html
I am trying to construct a Cayley table given these symmetry elements to show that the point group of this molecule is, in fact, a group (obeys closure property). This is how I wrote the group U, of the symmetry elements of the molecule:
Here is my attempt to construct a Cayley table for this. I realized this was wrong as C3*C2= C5/6, which is not part of the original set of symmetry elements, meaning the algebraic structure is not closed and therefore not a group.
I think the problem is a misinterpretation of the symmetry element. I look forward to your answer!