Easy character theory proof

Let u: G x S -> S and v: G x T -> T be two actions of a finite group G on two finite sets S and T. Let w: G x (S x T) -> (S x T) be defined by:

w(g,(x,y))=(u(g,x),v(g,y)) for all (x,y) in (S x T) and all g in G.

Let c1, c2, c3 be the characters of the permutation representations p1,p2,p attached to u,v,w respectively.

Prove that c1(g)=c2(g)c3(g) for any g.

So basically, prove the character of the action on the set (S x T) is equal to the product of the actions on the sets S and T.

For the trivial representation, this is obvious. 1=(1)(1)

But even for the sign representation, I don't get it. -1=/=(-1)(-1), for example. And then how to generalize to all representations of an abstract group G?

Thanks!