Group action of $C_n$ on $S_n$ Let $S_n$ be the $n$-th symmetric group and $C_n$ the $n$-th cyclic group (i.e. $C_n\cong\mathbb{Z}/n\mathbb{Z}$).
There is an action of $C_n$ on $S_n$ given by cyclically permuting any permutation (i.e. we think of $S_n$ as the set of all permutations of the ordered list $[1,\ldots,n]$, and then the action of $C_n$ is generated by $[i_1,i_2,\ldots,i_n]\mapsto[i_2,\ldots,i_n,i_1]$).
What, then, is the quotient $S_n/C_n$? It's clear that it must be of size $(n-1)!$, but beyond that, and explicit lists of elements for $n\leqslant4$, I am stuck when it comes to describing this set.
Ideally, I would like to understand the 'signed' quotient, i.e. the action on $S_n\times\mathbb{Z}/2\mathbb{Z}$ given by $([i_1,i_2,\ldots,i_n],0)\mapsto([i_2,i_3,\ldots,i_n,i_1], 1)\mapsto([i_3,\ldots,i_n,i_1,i_2],0)$ etc.
 A: You can obtain explicit representatives of this action as oriented necklaces labelled by $\{1,\ldots,n\}$. Concretely, take a permutation $\sigma = \sigma_1\cdots\sigma_n$ and consider the necklace oriented clockwise and labelled by the elements of $\sigma$ in that order. It is clear that to each necklace there corresponds $n$ permutations, and 'marking the number $1$' in a necklace gives representatives of the orbits. And for you second problem, I would consider coloured necklaces, $1$ or $-1$ in $C_2$ changing or not modifying the colour.
For example, representatives for $S_4$ are
$\require{AMScd}
\begin{CD}
\color{blue}{ \bf 1} @>>> 2\\
@AAA @VVV \\
4 @<<< 3 
\end{CD} $ $\require{AMScd}
\begin{CD}
\color{blue}{ \bf 1} @>>> 2\\
@AAA @VVV \\
3 @<<< 4 
\end{CD} $$\require{AMScd}
\begin{CD}
\color{blue}{ \bf 1} @>>> 3\\
@AAA @VVV \\
4 @<<< 2 
\end{CD} $$\require{AMScd}
\begin{CD}
\color{blue}{ \bf 1} @>>> 3\\
@AAA @VVV \\
4 @<<< 2 
\end{CD} $$\require{AMScd}
\begin{CD}
\color{blue}{ \bf 1} @>>> 4\\
@AAA @VVV \\
2 @<<< 3 
\end{CD} $$\require{AMScd}
\begin{CD}
\color{blue}{ \bf 1} @>>> 4\\
@AAA @VVV \\
3 @<<< 2 
\end{CD} $
A: Your action of $C_n$ on $S_n$ is by right multiplication, so $S_n/C_n$ is literally just the left coset space usually denoted $S_n/C_n$. Your action of $C_n$ on $S_n\times\Bbb Z_2$ is ill-defined unless $n$ is even, in which case it is still the left coset space $(S_n\times\mathbb{Z}_2)/C_n$ but with $C_n$ generated by $(\sigma,1)$ where $\sigma=(1\cdots n)$.
In the first case, since $C_n$ cycles the entries in a permutation's one-line notation, we can pick a representative of a coset by rotating any of its permutations until $n$ is the last entry. In other words, the usual copy of $S_{n-1}$ inside $S_n$.
More generally, if $G$ acts on a set $X$, $H$ is any regular subgroup and $K$ is a point-stabilizer then we have $G=HK$ and $H\cap K$ (so in particular $H$ and $K$ are representatives for $G/K$ and $G/H$ respectively, and $X\cong G/K$ as $G$-sets).
The same applies in the second case, but with $S_{n-1}\times\mathbb{Z}_2$.
