My question deals with the dihedral group $D_3$ of equilateral triangle 123 (1 top vertex, 2 bottom right vertex, 3 bottom left vertex).
R1 is the counterclockwise rotation of 120 degrees.
R2 is the counterclockwise rotation of 240 degrees.
SA is the symmetry through the top angle bissectrice (exchanging vertex 2 and 3)
SB is the symmetry through the right angle bissectrice (exchanging vertex 1 and 3)
SC is the symmetry through the left angle bisssectrice (exchanging vertex 1 and 2)
I constructed the Cayley table accordingly, and compared it to the $S_3$ Cayley table: they match very well, hence my deduction that:
R1 matches with permutation (123) of $S_3$.
R2 matches with permutation (321).
SA matches with permutation (23).
SB matches with permutation (12).
SC matches with permutation (31).
First question, is this correspondence fully right? The shapes of the two tables are quite similar, but I did not try all the possibilities.
Second problem :
I tried to deepen the above correspondence as follows :
If I write a symmetry in a line with 1st figure = top vertex, 2nd figure = right vertex and 3rd figure = left vertex, the symmetry R1 can be written