Find all the irreducible characters of $C_2\times C_3$. Note here that $C_n$ is a cyclic group of order n.
I have calculated the character tables for $C_2$ and $C_3$ and this is what I got:
$C_2$: $$\begin{array}{c|c|c|} & \text{E} & \text{$C_2$} & \text{Rotations} & \text{Quadratic}\\ \hline \text{A} & 1 & 1 & z, R_z & x^2, y^2, z^2, xy \\ \hline \text{B} & 1 & -1 & x, y, R_x, R_y & yz, xz\\ \hline \end{array}$$
$C_3$:$$\begin{array}{c|c|c|} & \text{E} & \text{$C_3$} & \text{($C_3$)$^2$} & \text{Rotations} & \text{Quadratic} & \text{Cubic functions}\\ \hline \text{A} & 1 & 1 & 1 & z, R_z & x^2 + y^2, z^2 & z^3, y(3x^2-y^2), x(x^2-3y^2), z(x^2+y^2) \\ \hline \text{E} & 1, 1 & \epsilon, \epsilon^* & \epsilon, \epsilon^* & x+iy; R_x + R_y, x-iy;R_x-iR_y& (x^2-y^2, xy), (yz, xz) & (xz^2, yz^2), [xyz, z(x^2-y^2)], [x(x^2+y^2), y(x^2+y^2)]\\ \hline \end{array}$$
I know that any representation of an abelian group has dimension $1$. $C_2 \times C_3$ will be abelian. Where would I go from here? Any help would be appreciated!