Is the order of the product of two commutating elements x, y; xy = yx, of the finite order of any group always equal to lcm (ord x, ord y)?

Question

Apparently $ord (xy)$ divides the least common multiple of $ord(x)$ and $ord(y)$. It can be easily proved that if $ord(x)$ and $ord(y)$ are coprime, then $$ord (xy) = ord(x).ord(y) = lcm (ord(x), ord(y)).$$ Must always be $$ord (xy) = lcm (ord(x), ord(y))?$$ If yes, can you prove it? If not, can you give a counterexample, i.e. find positive integers $ord(x), ord(y)$ with $gcd (ord(x), ord(y))\gt1$ such, that $ord (xy)\lt lcm(ord(x),ord(y))$?