Quaternionic representation of special linear group:SL(2,3)

Frobenius-Schur indicator
The Frobenius-Schur indicator of a representation is the inner product of the character of the representation and the indicator character (which is the character that assigns to every element is number of square roots). The Frobenius-Schur indicator can be computed as $$\sum \chi(g^2)$$ where $$\chi$$ is the character of the representation.

Note that for two elements in the same conjugacy class, their squares are also in the same conjugacy class as each other (though not necessarily the same conjugacy class as the original element). It thus suffices to compute $$\chi(g^2)$$ for one element in each conjugacy class and multiply by the size of the conjugacy class.

The Frobenius-Schur indicator is thus $$-24/24 = -1$$, indicating that the representation has a real character but cannot be realized over the field of real numbers.