Let $G=\langle(1,2,3),(1,2)(4,5,6,7)\rangle$ Find the center of $G.$
I find that $G=\langle x,y\mid x^3=y^4=e, xy=yx^2\rangle$. Can I use this presentation to find the center of the group? Is there a better strategy?
EDIT: By direct calculation and some guessing I find the center to be $y^2=(46)(57)$ but I cannot show this rigorously. Please help.