Let me first say that I'm a layman who's trying to understand group theory and gauge theory, so excuse me if my question doesn't make sense.
Before symmetry breaking, the Electroweak force has 4 degrees of freedom ($B^0$, $W^1$, $W^2$, and $W^3$ right?) and after symmetry breaking, we are left with the weak $SU(2)$ bosons ($W^+$, $W^-$, $Z$) and the photon. Why, then, is the symmetry group before Electroweak symmetry breaking $SU(2) \times U(1)$ and not just $U(2)$ (Since a Unitary group of $n$ dimension contains $n^2$ degrees of freedom?) What am I missing?