Find different symmetries of $D_4$ by composition of component symmetries.

Question

Need find the different symmetries of $D_4$ by composition.

Kindly tell if is the below process of deriving symmetries' permutations correct.

Let, $[1234]$ denote the initial configuration of square with top-left vertex being listed first. Here, it is denoted by $1,$ and the vertices are labelled in counter-clockwise direction from the top-left vertex.

Hence, the symmetries are:

1. $e=(1)(2)(3)(4),$ leading to the same configuration as the initial, i.e. $[1234],$
   
2. $r=(1234),$ leading to the configuration $[4123],$
   
3. $r^2=(13)(24),$ leading to the configuration $[3412],$
   
4. $r^3=(1432),$ leading to the configuration $[2341],$
   

Am assuming that $f$ denotes the reflection about the $x$-axis.

5. $f= (12)(34),$ leading to the configuration $[2143],$
   

For the rest, the configuration is found by the action of composition of symmetries, on the initial configuration of the labels of the vertices, $X.$ The labels are :$1,2,3,4;$ and are assumed to be distinct. The possible configurations of the four labels forms the set $X.$
Here, the symmetries define the possible configurations formed by the four distinct labels.
The symmetries form the permutations given by the group $D_4,$ while the set $X$ is made of (the symmetries as) configurations as members.

The initial configuration is taken as a set member $x_0\in X,$ and action on it, by symmetry $g_i\in G=D_4,$ is denoted by $g_i\times X\to X.$ The resulting configuration, in turn finds the permutation brought by the symmetry.

6. $rf= rf\times[1234]=r\times[2341]= [3214]=(13).$
   
7. $r^2f\times[1234]=r\times[3214]= [4321]=(14)(23).$
   
8. $r^3f\times[1234]=r\times[4321]= [1432]=(24).$

Answer 1

It seems fine to me. Well done!

However, once you have $r$ and $f$, you can compute the rest simply by composition of permutations. So your 6. to 8. are longer than they need to be.