Example of a group whose order is a perfect number with a "perfect" class equation

Question

I am interested in finding an example of a group G whose order is a perfect number and whose class equation is likewise "perfect". For example it could be a group of order 6 whose class equation looks like: 1 + 2 + 3. Thanks

Answer 1

As @Paul K points out, $S_3$ should be one. The conjugacy classes are $\{e\}$, $\{(123),(132)\}$ and $\{(12),(13),(23)\}$