$\operatorname{Sym}(P_3) = \bigl\langle r, f \mid r^3 = e = f^2, fr = r^{-1}f \bigr\rangle$. As $r^3 = e, r^{-1} = r^2$ hence $fr = r^2f$.
Relation 1: $fr = r^2f$, Relation 2: $fr = r^{-1}f$
Is $f^3r^4fr^2frf$ equal to $e$ or not?
Using Relation 1, I get $f^3r^4fr^2frf = r$.
Using Relation 2, I get $f^3r^4fr^2frf = e$.
This obviously can't be true, so where am I going wrong?