Size of conjugacy class need not divide exponent

Statement
It is possible to have a finite group $$G$$ and a conjugacy class $$c$$ in $$G$$ such that the size of $$c$$ does not divide the exponent of $$G$$.

This is a non-constraint on the fact about::conjugacy class size statistics of a finite group.

Opposite facts

 * Size of conjugacy class divides index of center
 * Size of conjugacy class is bounded by order of derived subgroup

Similar facts

 * Size of conjugacy class need not divide index of abelian normal subgroup
 * Size of conjugacy class need not divide order of derived subgroup

Example of the alternating group
In the alternating group of degree 4, the elements have orders 1,2,3, so the exponent is 6. However, there are two conjugacy classes of size 4, both comprising 3-cycles. 4 does not divide 6.