I am in the same situation. I am an introvert and dd is an extreme extrovert. Like your dd, my dd NEEDS to be with others. She NEEDS constant interaction. This can be difficult because I recharge by being alone and she recharges by being with others. I work hard at not making her feel I don't want to be with her all the time, but it's work. She's pushing me out of my comfort zone but it's well worth it. :)
She went to school K-3 and we're just ending our first year homeschooling. She misses seeing her friends daily, but we try really hard to keep her in touch with her established relationships from school (there were many so this is hard). We also keep her involved in outside activities (youth orchestra, piano, GS, community theater, dance, softball).
This morning, as she was reflecting on our year together, she asked me if I thought homeschooling was best for her. I deflected the question and asked for her opinion. Surprisingly, she said she loves homeschooling because she would never have the academic opportunities she has now and she wants to hs through high school. Even though she doesn't see her friends everyday she wouldn't change a thing. Honestly, I was pleasantly surprised.
Keep her home. She (and you) will find ways to keep her engaged.