Dog is submissive while alone but attacks strangers while in her "pack"

I have an 11-month-old mixed breed. She is very friendly and sociable with everyone. Submissive most of the time. Whenever she sees another dog while walking with us, she runs and says "hi".

A few months ago, we found a group of dogs with whom she "clicked", and ever since, we've met almost every night. The "pack" consists of 4 main members, with 5 or 6 more that come occassionally and play along.

With her "pack", she plays prefectly off-leash (all of them off-leash). We believe she's the "beta" member of the pack (another bitch is the alpha). However, recently, whenever another dog (a stranger) comes, she runs like she's gonna say "hi", but after a moment of reaching the other dog, she attacks. She doesn't "attack to kill", mind you, and she hasn't really hurt any of the dogs she's charged, but the other owners obviously react badly, and what if one day she really gets out of hand, or if the other dog attacks to kill?

How should I manage this? The times it's happened, I've slapped her and then forced her to a submissive posture, but by then the other owners have ran off, thus limiting the effect of my correction.

Thank you.