What is the minimal degree of a real faithful representation of the binary dihedral group $\mathrm{SL}(2,5)$?
Is there actually a unified way how to determine such minimal dimension (over the field of real numbers) for a finite group?
2026-04-22 02:04:11.1776823451
Real faithful representations of $\mathrm{SL}(2,5)$
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The answer to your first question is $4$. There is a $2$-dimensional faithful complex representation of ${\rm SL}(2,5)$ with real character values, but its representation is not realizable over the real field.
Its Schur Index is $2$, which means that the direct sum of two copies of this representation can be represented over ${\mathbb R}$.
I guess the answer to your second question is yes in the sense that there exist algorithms to compute the character table of a finite group, and also to compute Schur Indexes.