Number of irreducible representations over complex numbers with rational character values need not equal number of conjugacy classes of rational elements

Statement
It is possible to have a finite group $$G$$ such that the number oof irreducible representations of $$G$$ over the complex numbers with rational character values is not equal to the number of conjugacy classes in $$G$$ of rational elements.

Opposite facts
The result is true for some finite groups: Number of irreducible representations over complex numbers with rational character values equals number of conjugacy classes of rational elements for any finite group whose cyclotomic splitting field is a cyclic extension of the rationals

Here are some facts that are true for all finite groups:


 * Number of irreducible representations equals number of conjugacy classes
 * Number of irreducible representations over complex numbers with real character values equals number of conjugacy classes of real elements
 * Number of orbits of irreducible representations equals number of orbits under automorphism group
 * Number of irreducible representations over rationals equals number of equivalence classes under rational conjugacy
 * Number of irreducible representations over reals equals number of equivalence classes under real conjugacy