Defining dihedral groups $\{\sigma \in S_n: $ something $\}$ I am trying to understand hos one can define the dihedral groups $D_n$. I have seen the "definition" that just says this is the group of symmetries of an $n$-polygon. So you have rotations and reflections. But I feel this definition is a bit vague. I asked around and heard that one can define this using generators and relations. I don't know about that.
I know that one can "realize" for example $D_4$ as a subgroup of $S_4$. For example
$$
D_4 =\{(1), (13), (24), (14)(23), (1234), (12)(34), (1432), (13)(24)\}.
$$
I do understand that I get these elements from labelling the vertices of the $4$-gon. This is very concrete for me.
Therefore my question is: Is there a nice way to actually define the general dihedral group $D_n$ as a specific concrete subgroup of $S_n$?
So, for example, I am looking for something like 
$$D_n = \{\sigma \in S_n : \text{something} \}.$$
I am not looking for a vague algorithmic way of defining the groups.
From the example of $D_4$ I am thinking that it should always have $2$-cycles, but I don't think that this is always true. For example with $D_5$.

Yes.  Label the vertices of your regular $n$-gon with $1, 2, 3, \ldots, n$.  Keep track of where the vertices go after the rotation or reflection is done.  The result is a literal permutation on $\{1, 2, 3, \ldots, n\}$ and the operation is still composition, so $D_n$ is a subgroup of $S_n$.
