dihedral group

The dihedral group $D_{n}$ is the group of symmetries of a regular $n$ -gon under composition.

The order of $D_{n}$ is $2 n$ :

$n$ rotational symmetries
$n$ reflectional symmetries

Example: $D_{4}$ :

\documentclass[tikz,margin={4cm 0cm}]{standalone}
%
\definecolor{accentcolor}{HTML}{8A5CF5}
%
\newcommand{\square}{%
    \draw (45:1) -- (135:1) -- (225:1) -- (315:1) -- cycle;
}
%
\newcommand{\rotarc}[2]{%
	\begin{scope}
		\clip (#1) rectangle (#2);
		\draw[dashed,accentcolor,very thick] (0,0) circle (1);
	\end{scope}
}
%
\begin{document}
\begin{tikzpicture}
    \begin{scope}
        \square
	    \rotarc{0,0}{2,2}
        \node at (-2,0) {$R_{90}$};
    \end{scope}
    \begin{scope}[yshift=-3cm]
        \square
        \rotarc{-2,0}{2,2}
        \node at (-2,0) {$R_{180}$};
    \end{scope}
    \begin{scope}[yshift=-6cm]
        \square
        \rotarc{-2,0}{2,2}
        \rotarc{-2,-2}{0,0}
        \node at (-2,0) {$R_{270}$};
    \end{scope}
    \begin{scope}[yshift=-9cm]
        \square
        \rotarc{-2,-2}{2,2}
        \node at (-2,0) {$R_{360}$};
    \end{scope}
    \begin{scope}[xshift=7cm]
        \square
        \draw[dashed,accentcolor,very thick] (-1.1,0) -- (1.1,0);
        \node at (-2,0) {$H$};
    \end{scope}
    \begin{scope}[xshift=7cm,yshift=-3cm]
        \square
        \draw[dashed,accentcolor,very thick] (0,-1.1) -- (0,1.1);
        \node at (-2,0) {$V$};
    \end{scope}
    \begin{scope}[xshift=7cm,yshift=-6cm]
        \square
        \draw[dashed,accentcolor,very thick] (-1.1,1.1) -- (1.1,-1.1);
        \node at (-2,0) {$D$};
    \end{scope}
    \begin{scope}[xshift=7cm,yshift=-9cm]
        \square
        \draw[dashed,accentcolor,very thick] (-1.1,-1.1) -- (1.1,1.1);
        \node at (-2,0) {$D'$};
    \end{scope}
\end{tikzpicture}
\end{document}

Its Cayley table is

	$R_{90}$	$R_{180}$	$R_{270}$	$R_{360}$	$H$	$V$	$D$	$D^{'}$
$R_{90}$	$R_{180}$	$R_{270}$	$R_{360}$	$R_{90}$	$D$	$D^{'}$	$V$	$H$
$R_{180}$	$R_{270}$	$R_{360}$	$R_{90}$	$R_{180}$	$V$	$H$	$D^{'}$	$D$
$R_{270}$	$R_{360}$	$R_{90}$	$R_{180}$	$R_{270}$	$D^{'}$	$D$	$H$	$V$
$R_{360}$	$R_{90}$	$R_{180}$	$R_{270}$	$R_{360}$	$H$	$V$	$D$	$D^{'}$
$H$	$D^{'}$	$V$	$D$	$H$	$R_{360}$	$R_{180}$	$R_{270}$	$R_{90}$
$V$	$D$	$H$	$D^{'}$	$V$	$R_{180}$	$R_{360}$	$R_{90}$	$R_{270}$
$D$	$H$	$D^{'}$	$V$	$D$	$R_{90}$	$R_{270}$	$R_{360}$	$R_{180}$
$D^{'}$	$V$	$D$	$H$	$D^{'}$	$R_{270}$	$R_{90}$	$R_{180}$	$R_{360}$