Field generated by character values need not be cyclotomic

Statement
It is possible to have a finite group $$G$$ such that the field generated by character values in characteristic zero is not a cyclotomic extension of the field of rational numbers.

Similar facts

 * Minimal splitting field need not be cyclotomic
 * Minimal splitting field need not be contained in a cyclotomic extension of rationals

Opposite facts

 * Field generated by character values is contained in a cyclotomic extension of rationals

Other related facts

 * Minimal splitting field need not be unique
 * Splitting not implies sufficiently large
 * Field generated by character values is splitting field implies it is the unique minimal splitting field

Examples where the field generated by character values is the unique minimal splitting field
There are many examples among dihedral groups. The minimal splitting field for a dihedral group of degree $$n$$ and order $$2n$$ coincides with the field generated by character values. It is $$\mathbb{Q}(\cos(2\pi/n))$$, which is a subfield of the reals. When $$n \ne 1,2,3,4,6$$, then this is strictly bigger than $$\mathbb{Q}$$, and hence is not a cyclotomic extension of $$\mathbb{Q}$$.

Here are some examples (including dihedral groups and others):