TITLE: Name and role of a particular finite group?
QUESTION [6 upvotes]: The group generated by the functions $x\mapsto 1/x$ and $x\mapsto 1-x$ with composition of functions as the group operation is a non-abelian group with only six elements (listed below).
Does this particular realization of $S_3$ have a conventional name?  Is there a literature concerning it?  Does it interface in interesting ways with geometry, probability, combinatorics, algebra, number theory,${}\,\ldots\,{}$?
\begin{array}{rcl|c|cccc}
& & & \text{order} & \text{fixed points} \\
\hline
x & \mapsto & x & 1 & \mathbb C\cup\{\infty\} \\[10pt]
x & \mapsto & 1/x & 2 & \pm 1 \\[6pt]
x & \mapsto & 1-x & 2 & 1/2,\ \infty \\[6pt]
x & \mapsto & x/(x-1) & 2 & 2,\ \infty \\[10pt]
x & \mapsto & (x-1)/x & 3 & (1\pm i\sqrt 3)/2 \\[6pt]
x & \mapsto & 1/(1-x) & 3 & (1\pm i\sqrt 3)/2
\end{array}
PS: Since this is isomorphic to the group of permutations of three elements, one could wonder which three elements it permutes, and it seems we should take those to be $0$, $1$, and $\infty$.
The first element of order $2$ transposes $0$ and $\infty$ while leaving $1$ fixed.  The second transposes $0$ and $1$ while leaving $\infty$ fixed.  The third transposes $1$ and $\infty$ while leaving $0$ fixed.
The first element of order $3$ takes $0$ to $\infty$, $\infty$ to $1$, and $1$ to $0$.  The second takes $0$ to $1$, $1$ to $\infty$, and $\infty$ to $0$.
PPS: Evaluate the derivative of each of these at one of the fixed points.  Unsurprisingly, you get a primitive $n$th root of $1$, where $n\in\{1,2,3\}$ is the order.  The fixed point is the axis of a rotation of $0^\circ$, $180^\circ$, or $120^\circ$.

REPLY [4 votes]: This particular representation of $S_3$ appears in the theory of elliptic curves, specifying the fractional linear transformations that can be applied to a parameter $\lambda \in \mathbb{C} \setminus \{ 0, 1 \}$ that do not change the isomorphism class of the elliptic curve (in Legendre normal form).
$$y^2 = x(x - 1)(x - \lambda).$$
See this blog post for some details. The value of $\lambda$ itself, separate from its orbit under this action, describes an elliptic curve with level structure where $N = 2$. The orbit of $\lambda$ under the action also describes the possible cross ratios of the four $2$-torsion points of the elliptic curve, namely $0, 1, \infty$, and $\lambda$.