Give an example of commuting elements $x,y$ such that the order of $xy$ is not equal to the least common multiple of $|x|$ and $|y|$

Question

Assume $|x|=n$ and $|y|=m$ . Give an example of commuting elements $x,y$ such that the order of $xy$ is not equal to the least common multiple of $|x|$ and $|y|$

My attempt : I was thinking about taking the non-abelian Group $S_3$.Take $x=(12)$and $y=(123)$ but here $xy \neq yx $ i,e $x$ and $y$ are not commuting elements

After that I'm thinking about again abelian group $K_4$ where $K_4=\{1,x,y,xy\} $

Here $|x|=2$ and $|y|=2$ .So $Lcm(x,y)=lcm(2,2)=2$ but $|x||y|=|xy|=2.2=4 $

I think in $K_4$, order of $xy$ is not equal to the least common multiple of $|x|$ and $|y|$

Answer 1

Take a non-identity $x$ and $y=x^{-1}$, then $|xy|=1 \neq lcm(|x|,|y|)=|x|$.