How to find the character table of a group with 12 elements generated by $x$ and $y$ with $x^3=y^2$

Question

I need to find the character table of the group with 12 elements generated by $x$ (of order 6) and $y$ (of order 4) with $x^3 = y^2$ and $y^{-1}xy = x^{-1}$. I have elements $e, x, y, xy, x^2y, x^2$ but as the question says there are 12 elements. How would you then make a character table out of this?

Answer 1

Elements are those 6 plus $x^3,x^4,x^5,x^3y,x^4y,x^5y$. The relation $y^{-1}xy=x^-1$ gives $yx=x^{-1}y=x^5y$.

You can use that (and the other relations) to get the product of any two elements. For example, $(x^2y)(xy)=x^2(yx)y=x^2x^{-1}y^2=xy^2=x^4$.