= Permutation order and sign do not determine conjugacy
<Conjugation> preserves <order of a group element>, because $(gxg^{-1})^k=gx^kg^{-1}$, and preserves the <sign of a permutation>, because sign is a <group homomorphism> into $\{1,-1\}$. These invariants do not determine <cycle type>. In $S_6$, $(123)$ and $(123)(456)$ both have order three and positive sign, but have three and zero fixed points respectively. Conjugation relabels cycles, so these <permutations> are not conjugate.
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