The Cyclic Groups $(\mathbb Z_n,+)$ have various representations. This answer asserts that the only irreducible representations are
- 1-dimensional, with matrix a real $n$th root of unity;
- 2-dimensional, with matrix $\left(\begin{smallmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{smallmatrix}\right)$ where $n\theta \equiv 0 \pmod{2\pi}$ but $\sin\theta \neq 0$.
But what about other minimal unitary matrix representations?
Take as an example $\mathbb Z_6$, which has other representations. Such as the 6 cyclic permutations of
$$\pmatrix{1&0&0&0&0&0\\0&1&0&0&0&0\\0&0&1&0&0&0\\0&0&0&1&0&0\\0&0&0&0&1&0\\0&0&0&0&0&1} $$
constructed by repeatedly moving the leftmost column to the righthand side of the matrix and shifting all of the other columns left by 1 space.
Or the representation of 6 5x5 matrices constructed from the direct sum of the $\mathbb Z_2$ and $\mathbb Z_3$ representations
$$\left\{\pmatrix{1&0\\0&1},\pmatrix{0&1\\1&0}\right\} $$ and $$\left\{\pmatrix{1&0&0\\0&1&0\\0&0&1},\pmatrix{0&0&1\\1&0&0\\0&1&0},\pmatrix{0&1&0\\0&0&1\\1&0&0}\right\} $$ respectively.
Neither of these representations can be written as a direct sum of other representations of $\mathbb Z_6$. In other words, as far as I can tell, these representations have only trivial subrepresentations; therefore, why should they not be considered irreducible (as per the linked question)?
With $\mathbb{Z}_6$ acting on $\mathbb{R}^6$ by cyclic coordinate shifts, the irreducible subrepresentations are
With $\mathbb{Z}_6\cong\mathbb{Z}_2\times\mathbb{Z}_3$ acting on $\mathbb{R}^2\oplus\mathbb{R}^3$ by cyclic coordinate shifts in each component, they are