I'm using GAP software to find irreducible representations of certain groups, but in many cases the representations are complex and what I need is real representations.
My question is more precisely, given a group $G$ with linear irreducible representation $\rho:G \rightarrow GL_n(\mathbb{C})$ with complex entries in some $\rho(g), g \in G$, is there a straightforward way to obtain a real irreducible representation $\sigma: G \rightarrow GL_m(\mathbb{R})$ corresponding to $\rho$?
I would also be happy with just a computational way of obtaining the representations or a look-up table/resource for usual small groups (order $\leq$ 100).
Thank you in advance.
Consider the group $C_3$ of three elements. It has exactly three irreducible representations, the trivial one and two that involve nonreal cube roots of unity. You can find real representations of this group (the regular representation is always real, for eaxmple), but other than the trivial one they can't be irreducible.
EDIT: OK, we want real representations that are irreducible over the reals, not necessarily irreducible over the complex numbers. I don't know how to do that for finite groups in general, but for groups of order at most six, here's how:
For the groups of orders 1 and 2, and also for Klein-four and $S_3$, the complex irreducibles are real-valued, so there's nothing to do. The other groups are groups of rotations, so there are representations involving $$\pmatrix{a&-b\cr b&a\cr}$$ where $a$ and $b$ are the sines and cosines of the appropriate angles.