$D:= \mathbb{R} \setminus\{0,1\}$ $f(x)= \frac{1}{x} ,g(x)=\frac{x-1}{x} $
To show that the group $G$ generated by those two functions (with $\circ $) is isomorphic to the group $S_3$
I found out that the group is $G=\{x, \frac{1}{x}, \frac{x-1}{x},\frac{1}{1-x}, \frac{x}{x-1}, 1-x\}$ (it has $6$ elements)
I need to show that with linking Panels. Should look like linking panel of $S_3$.
But I can't find the right order of $G$. Hope someone can help :)
Your group has one element of order $1$ ($x$), three elements of order $2$ ($\frac1x$, $1-x$, and $\frac x{x-1}$) and two elements of order $3$ ($\frac{x-1}x$ and $\frac1{1-x}$). Now, simply compute the composition of any two elements of $G$ and you will get basically the same table that you get with the group $S_3$.