The small dihedral groups $D_1$ and $D_2$. In M. Artin's book Algebra he wrote:


But I think this visualisation $D_1$ and $D_2$ is inconsistent and confusing, because I guess $D_n$ could be associated with a subgroup of $S(\{1, \ldots, n\})$, i.e. permutations of the $n$ vertices, but in the case of the $1$-gon there is just one permutation, the identity, and in the case of the $2$-gon there are just two (identity and one transposition). Furthermore by a reflection of the $2$-gon in the horizontal axis it gets projected onto itself (every vertex stays where it is), so this reflection is actually the identity, and not a separate element $r \ne 1$. Or does I miss something? Could it be consistent to look at $D_1$ and $D_2$ in this way?
What Artin is saying is excactly that


  because I guess $D_n$ could be associated with a subgroup of $S(\{1,\ldots,n\})$, i.e. permutations of the $n$ vertices


is not true for $D_1$ and $D_2$ (at least with the definition he uses for that notation).

By the symmetry of the $n$-gon you should, in this case, think of the group of all the isometries of the plane that fix the outline of the $n$-gon. For $n=1$ there are two such isometries, namely the identity and the reflection through the midline.

It's a general fact about isometries of the plane that knowing what they do to three non-collinear points is enough to reconstruct the entire isometry. In particular for $n\ge 3$ it just happens that knowing what happens to each vertex of an $n$-gon is enough to reconstruct an entire isometry, so in that particular case one can represent elements of $D_n$ by permutations from $S_n$. But that's just a practical property that holds in the case $n\ge 3$, not part of the definition of $D_n$.
1 and 2 are the only positive integers $n$ for which $2n \nmid n!$, which is why $D_2$ and $D_4$ are not subgroups of $S_1$ and $S_2$.