I am curious about the irreducible representations $\rho: D_{2p} \rightarrow GL_n(\mathbb{Q})$ of dimension at most $p-1$, not the real or complex representations. My mind is occupied with these two questions, especially the first question:
Given a prime number $p$, what are the faithful irreducible representations of the dihedral group $D_{2p}$? What are the irreducible representations of the dihedral group $D_{2p}$?
For $p=5$, do we have irreducible faithful representations of dimensions less than $4$? How can we write down the $4\times4$ rational matrices corresponding to $D_{10}$?