I don't think you are being too strict. I think going in a group might be okay. Lots of kids at the movies, picked up right after the movie by a parent might be acceptable.
My kids started dating at age 16 but spent a lot of time here with their dates, watching movies and things like that. I know kids think that is it a matter of us not trusting them - but that isn't it at all. As adults we are so aware of things that can happen when kids are left alone. They simply aren't equipped with the emotional maturity to handle some things that might come up--and I don't just mean sexual things.
Also remember if you give in and let her do this now, it sets a precedent and there is no going back. You simply cannot unring that bell.