Is Dihedral group just for small order? I'm sorry that the title is not really specifying a question, but i cannot think of a single sentence describing this question.
Dihedral group is defined to be rotations and reflections of $n$-agon in elementary abstract algebra texts. These groups are good examples for nonabelian groups.
For very small cardinals such as $6,8,10$, we can actually construct a dihedral group formally by checking a diagram.
However, for an arbitrary $N$, we cannot explicitly do this and the definition above is not formal at all.
Am i thinking correctly? If not, what would be a way to define Diheral group formally?

There is a formal construction of the dihedral group for arbitrary $n$. Here is my favorite way to do it: let $r$ denote the rotation which moves the vertices of an $n$-gon by $2\pi/n$ radians and let $s$ be reflection about a fixed line. The elements $r$ and $s$ satisfy the properties
\begin{align}
r^n &= 1 \\
s^2 &= 1 \\
srs &= r^{-1}.
\end{align}
It turns out that these three properties completely characterize the dihedral group of order $2n$. In other words, I have given you a presentation for the dihedral group.
