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Attempt:
(1S,3S)-1,3-Dichlorocyclohexane
I'm just not sure about the nomenclature for R and S configurations (our teacher was a little unclear). I gave it my best shot after looking through some textbooks but would like just a quick confirmation or someone to tell me if it's wrong.
In addition, if I exchanged the positions of the Cl and H groups on each side with each other would I get an enantiomer? ((1R,3R)-1,3-Dichlorocyclohexane)
It's correct, as already mentioned in a comment.
Some might find more or less helpful to redraw the structure to:
Your additional question can be tricky, understood a different way. It's possible to exchange H and Cl positions, or rather axial to equatorial bonds and the other way around – by switching to the other boat conformation, i.e. without changing the configuration:
(The two structures are identical, one of them can be rotated to superimpose the other.)
