Character table of $G=\langle (12),(34),(45)\rangle$.

Question

I consider the group $S_5$ and the subgroup $G=\langle (12),(34),(45)\rangle$. With help, I could find the conjugacy classes of $G$ and now I want to find the characters. Using that $\vert G \vert$ equals the sum of the squares of the $\chi(1)$'s I can fill in the first column of the table. Also, Since I have the trivial and the sign representation, I can fill the first and second row. Now since (34) and (45) are in the same conjugacy class, I have $\lambda_i(34)=\lambda_i(45)$, so $\lambda_i((345))=\lambda_i((34)(45)) = \lambda_i(34)^2=1$ since because $(34)$ is of degree $2$. Now this is as far as I can go. What trick can I use to continue and how? \begin{array}{rrrrrrrrrrr} & (1) & (34) & (345) & (12) & (12)(34) & (12)(345)\\ \text{Triv}& 1 & 1 & 1 & 1 & 1 & 1\\ \text{Sign}&1&-1&1&-1&1&-1\\ \lambda_1&1&&1&&&\\ \lambda_2&1&&1&&&\\ \chi_1&2&&&&&\\ \chi_2&2&&&&& \end{array}

Answer 1

The group has a quotient $S_3$ (acting on the points 3..5). Do you know the character table of $S_3$? Do you know how to lift a chjaracter from a quotient group to $G$? This gives you a character of degree 1 and one of degree 2. Tensoring with the sign then yields the full atble.