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I am studying the dihedral group and comparing the table I have in my book with that of wikipedia. I realize that the entries were inverted, that is XY in the wikipedia article corrisponds to YX in my book.

Let $D_3=\{1,R1,R2,S1,S2,S3\}$ be the dihedral group of the triangle- with $R1$ the counter-clockwise rotation of 120 degrees and $R2$ the the counter-clockwise rotation of 240 degrees. $Si$ is the reflection about the i-the symmetry axis(that is the axis from the vertex i to the midpoint of the opposite side) $i=1,2,3$.

I guess a possible explanation is that the definition of the symmetry axes is ambiguos:

If the axes of symmetry aren't fixed, they move with the vertices. What I mean is that for example: If I draw a triangle and number the vertices, initially the symmetry axis r2 corresponding to the vertex 2 has a certain position. In the operation $S3S2$ (first $S3$ and then $S2$): $S3$ changes the position of the vertex 2 and so does r2, then $S2$ will be the reflection with respect to the new position of the axis r2. So $S3S2=R2$.

If the axes are fixed geometrical objects,then in $S3S2$ then$ S3$ changes the position of the vertex 2, but the axis labelled r2 is still the one at the beginnig, because it doesn't move. Then $S2 $will be the reflection with respect to the initial position of the axis r2. This yields $S3S2=R1$.

So how should reflections be interpreted and what comes first in XY: X or Y?

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    $\begingroup$ We can always relabel, and $XY$ is not uniquely defined. For me, as maps, it is $(XY)(i)=X(Y(i))$, so $Y$ first. $\endgroup$ Jun 22 '20 at 10:50
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Yes, the axes of symmetries are fixed and your explanation is correct.

Regarding what comes first, most authors stick to the convention of right to left like in function composition, this has become a standard. But to remove ambiguity for the reader, the author would mention the convention he/she would be following in their book, since they know a different convention exist.

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