Schur index of irreducible character

Direct definition
Suppose $$G$$ is a finite group, $$K$$ is a defining ingredient::splitting field for $$G$$, and $$\chi$$ is the character of an defining ingredient::irreducible linear representation $$\varphi$$ of $$G$$ over $$K$$. Suppose $$k$$ is the subfield of $$K$$ generated by the character values $$\chi(g), g \in G$$. The Schur index of $$\chi$$ (also termed the Schur index of $$\varphi$$) is defined in the following equivalent ways:


 * 1) It is the smallest positive integer $$m$$ such that there exists a degree $$m$$ extension $$L$$ of $$k$$ such that $$\varphi$$ can be realized over $$L$$, i.e., we can change basis so that all the matrix entries are from $$L$$. Note that it is not necessary that $$L$$ be a subfield of $$K$$, but rather we need to work within a suitable larger field that contains both $$L$$ and $$K$$ to perform the necessary conjugation.
 * 2) It is the multiplicity of $$\varphi$$ in any irreducible linear representation $$\alpha$$ of $$G$$ over $$k$$ that has $$\varphi$$ as one of its irreducible constituents over $$K$$.

The Schur index of a character $$\chi$$ is often denoted $$m(\chi)$$.

Note that if the representation can be realized over the field generated by the character values for that representation, the Schur index is one.

Facts

 * Schur index of irreducible character need not equal degree of extension of minimal field realizing the character over field generated by character values
 * Schur index of irreducible character is one in any prime characteristic
 * Schur index divides degree of irreducible representation
 * Schur index of irreducible character in characteristic zero divides exponent
 * Square of Schur index of irreducible character in characteristic zero divides order
 * Odd-order p-group implies every irreducible representation has Schur index one

Examples
From the facts above, it is clear that to get an example of an irreducible character/representation with Schur index greater than 1, we should not look at odd-order $$p$$-groups. Here are the smallest examples:


 * The smallest example with Schur index two is faithful irreducible representation of quaternion group. The field generated by character values is $$\mathbb{Q}$$. The representation cannot be realized over $$\mathbb{Q}$$ (this follows from the indicator theorem) but it can be realized in any quadratic extension of the form $$\mathbb{Q}(\sqrt{-m^2 - 1})$$ for $$m \in \mathbb{Q}$$.
 * The smallest example with Schur index three is a representation of the nontrivial semidirect product of Z7 and Z9 (order 63). Note that we cannot find any examples of Schur index three using 3-groups, because odd-order p-group implies every irreducible representation has Schur index one.