What is the best way to look at the symmetric group $S_3$in relation to a triangle?

Question

For the group $S_3$, we have 6 elements or functions(the upper row is the input, the lower row is the output:

$\rho_0=\begin{pmatrix}1&2&3 \\ 1&2&3\end{pmatrix},\ \rho_1=\begin{pmatrix}1&2&3 \\ 2&3&1\end{pmatrix},\ \rho_2=\begin{pmatrix}1&2&3 \\ 3&1&2\end{pmatrix} \\ \mu_1=\begin{pmatrix}1&2&3 \\ 1&3&2\end{pmatrix},\ \mu_2=\begin{pmatrix}1&2&3 \\ 3&2&1\end{pmatrix},\ \mu_3= \begin{pmatrix}1&2&3 \\ 2&1&3\end{pmatrix}$

It is mention that the $\rho's$ are rotations, and that the $\mu's$ are mirror translations.

But I get a little confused because there are basically two ways of looking at it, wheter we do the rotations clockwise or counterclockwise.

Lets just look at $\rho_1$. If we start with rotating clockwise we get:

Here we get that the order 2,3,1 comes in the same order as 1,2,3 as in the original triangel. But now we have to look at the original position as function inputs, and the numbers as outputs. That is the original "point 1" is transformed to point 2, the original point 2 is transformed to point 3, the original point3 is transformed to point 1.

But look at the counter-clockwise rotation:

Now the numbers are the functions-input and the point in the plane is the output. So the number 1 is sent to the point 2, the number 2 is sent to the original point 3, and the number 3 is sent to the original point 1.

My questions: Which convention is usually used? My book only says that this gives a the original six functions gives a correspondence to the symmetric groups of an equileteral triangle. But what precisely is meant with this? Is there a usual convention that always is used, or are both used? Is one better to use? My book doesn't state if $\rho_1$ corresponds to a clockwise, or counter-clockwise rotation.

Answer 1

Once you fix the numbering of the vertices of the triangle, the two elements of order $3$ of the group correspond to two rotations, one being the inverse of the other one. There is no canonical choice a priori, and both are good. This gives two different isomorphisms between $S_3$ and the group of isometries of the triangle, which differ by a conjugation.

Answer 2

The difference in these two actions is in which side they act on. If you consider the functions to permute the positions, then the action is on the right, meaning for the product $fg$ the function $f$ is applied first. If you consider the functions as having the numbers as inputs, then $g$ would be applied first. These actions differ by inversion; that is, you can turn the right action into a left action by applying the inverse map. This means $fg$ acting on the right would become $g^{-1}f^{-1}$ acting on the left. Essentially the actions are therefore the same.