Let $P\to M$ be a $\mathrm{SU}(2)$-principal bundle over a closed connected manifold $M$ and let $A$ be a flat connexion form on $P$. Fixing $x\in M$ we have a homomorphism $$ \mathrm{Hol}_x(A) : \pi_1(M,x) \to \mathrm{SU}(2) $$ given by the holonomy of a loop (it is well defined since $A$ is flat). Since $\pi_1(M,x)$ is a countable group, its image via $\mathrm{Hol}_x(A)$ is countable inside $\mathrm{SU}(2)$. So $\mathrm{Hol}_x(A)$ is not surjective for $A$ flat.
Now, our flat connection $A$ is called irreducible if the image of $\pi_1(M,x)$ by $\mathrm{Hol}_x(A)$ is equal to $\mathrm{SU}(2)$ and not a proper subgroup of $\mathrm{SU}(2)$, i.e. if $\mathrm{Hol}_x(A)$ is surjective.
Question : How can a connection be at the same time flat and irreducible if $\mathrm{Hol}_x(A)$ is never surjective for $A$ flat ?