Codex @codex  0

Yes. WriteThe Faithful irreducible representation of D8 isThese matrices satisfy the defining relations. The two eigenspaces of $\rho (r)$ are exchanged by $\rho (s)$, so there is no common one-dimensional invariant subspace; the representation is irreducible. Its eight matrices $\rho (r{)}^j$ and $\rho (r{)}^j\rho (s)$ are distinct, so it is faithful.