Character table of S4

Question

I am trying to understand the character table of $S_4$. I have obtained the trivial, signature and standard representations. The fourth one is the product of signature and standard.

Now for the last representation, I have tried two things.

1. I took the standard basis for $C$, defined $\pi(\sigma)e_i = e_{\sigma i}$. I took the trace of the matrices as characters, but there it turned out that $\pi$ is just direct sum of trivial and standard representations.

2. I have read that I should take the normal subgroup corresponding to (12)(34). I don't know how to go about it. Can somebody explain? Any other method to determine the last representation is also welcome.

Thanks in advance.