Is the subgroup $H:=\langle xyxy\rangle$ normal in $G:=\langle x,y\mid x^3=e, y^2=e\rangle$? If so, find $G/H$.

Question

Let $G=\langle x,y\mid x^3=e, y^2=e\rangle$ and otherwise unrestricted.

Let $H=\langle xyxy\rangle$.

Is $H\triangleleft G$?

And in that case, what is $$G/H$$ isomorphic to?

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If it is a normal subgroup, I'd suspect $G/H\cong D_3$, but is that really true?