What is the smallest poset with automorphism group $C_n$?

I've recently been interested in finding small finite posets (and thereby finite $T_0$ topologies) with a given automorphism group.

I came upon the paper of Barmak and Minian in which they provide an upper bound for this problem. In particular, for any group $G$ they construct a poset on $n(r+2)$ points with automorphism group $G$, where $n = |G|$ and $r$ is the size of a generating set for $G$.

It seemed clear that in many cases, especially the Abelian case, this bound was rather weak. The bound for cyclic groups is $3|G|$, but this can be improved significantly whenever $G$ can be expressed as a direct product $H \times K$ by joining a poset for $H$ and a poset for $K$ together and letting all points of one poset be below all points of the other. For $C_{15}$ this gives a bound of $24$ points as opposed to $45$.

I also came across results about the problem for graphs instead of posets and it seems much is known in the Abelian case, as is detailed in Woodruff's survey of graphs of minimum order with given automorphism group. In fact, in the case of graphs the bound can be improved for many cyclic groups which aren't direct products. For example, for $C_4$ a graph with $10$ vertices is minimal as opposed to $12$, as can be seen in this question: An example of a simple graph whose automorphism group is isomorphic to the cyclic group on 4 elements.

Given that graphs can encode more structure than posets, it appears that the minimum sizes for graphs with a given automorphism group provide lower bounds for the minimum sizes for posets with a given automorphism group. I was hoping that in most cases I'd be able to translate a minimum graph with given cyclic automorphism group into a minimum poset with that automorphism group, but I can't even seem to make this work for $C_4$. The smallest poset I can find for $C_4$ has $12$ points. So perhaps it's not so straightforward and the case of posets is significantly different than that of graphs.

What is known about the minimum poset with automorphism group $C_n$? I'd probably settle for knowing the minimum poset for $C_4$.

For $C_2$ the minimum poset is of course the 2-element antichain.

By direct inspection of all posets of at most 12 elements (slightly more than 1 billion posets, code for generating them can be found here), we find that:

$C_3$ first appears in a poset of 9 elements. There is one such poset:

$C_4$ first appears in posets of 12 elements. There are 7 such posets:

$C_5$ does not appear in any poset with 12 elements or less, and neither do any $C_n$ for $n \ge 7$.

Interestingly, $C_6$ first appears already in posets of 11 elements, with 7 cases. The first case corresponds to the OP's construction (joining a 2-element poset for $C_2$ on top of a 9-element poset for $C_3$).