Field generated by character values need not be a splitting field

In characteristic zero
It is possible to have a finite group such that the fact about::field generated by character values in characteristic zero is not a fact about::splitting field.

Related facts

 * Field generated by character values is splitting field implies it is the unique minimal splitting field
 * Rational not implies rational-representation

Proof
The idea is to pick a group where one or more of the representations have Schur index greater than 1. The smallest example is the quaternion group, where the field generated by character values is the field of rational numbers (so the group is a rational group) but the two-dimensional faithful irreducible representation cannot be realized over the rationals or even over the reals (this can be shown via the indicator theorem).