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 Conjugacy class size statistics of a finite group (?).

Related facts

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

Proof

Example of the alternating group

Further information: alternating group:A4, element structure of alternating group:A4

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.