Rational-representation group

Definition
A rational-representation group is a finite group satisfying the following properties:


 * 1) The field of rational numbers $$\mathbb{Q}$$ is a splitting field, i.e., every irreducible representation in characteristic zero is realizable over the rational numbers.
 * 2) Every irreducible representation in characteristic zero can be realized over $$\mathbb{Z}$$.

Equivalence of definitions
Definitions (1) and (2) are equivalent because linear representation is realizable over principal ideal domain iff it is realizable over field of fractions.

Examples
Here are some small examples, with relevant contextual information.

Facts

 * Symmetric groups on finite sets are rational-representation groups.
 * Dihedral group:D8 is a rational-representation group.
 * Trivial group, cyclic group:Z2, and dihedral group:D8 are the only three groups with the property that all their irreducible representations can be written over the rationals as orthogonal matrices.

Weaker properties

 * Stronger than::Rational group: A rational group is a finite group such that all its characters are rational-valued (hence integer-valued). The quaternion group is an example of a rational group that is not a rational-representation group.
 * Stronger than::Ambivalent group: A finite group in which every character is real-valued.