Is $U(1)\times SU(2) \times SU(3)$ a vector space over a field? I saw an article [here](http://en.wikipedia.org/wiki/Group_extension) that seemed to me that a similar concept to a field extension was being used. 

In QFT, is each particle considered to be its own vector space? 

And are the individual vector spaces then bridged together by these group extensions?

If I have asked too many questions, feel free to respond to just one.