I've been here with my DD (15 this week), OP. For some reason, she sometimes gets it into her head to be rude, show off, obnoxious when she has a friend here. Well, she can be all of those things anyway, but it seems even worse sometimes when she thinks she has a sympathetic audience.
I have sent friends home for this kind of behaviour. However, I have done it after a very clear warning. In the warning, I specified which aspects of DD's behaviour I was objecting to, and which things were rude (though it can be very difficult to put your finger on exactly what is rude, when it's a generally piss-taking and obstreperous attitude, rather than any one specific word or phrase).
The one time I sent DD's friend home, DD was absolutely enraged. Said her friend now saw how unreasonable I was, she (DD) wasn't rude, didn't know what she'd done wrong, etc, etc, etc, etc. I just said very calmly that I warned her. I have also refused to let that friend sleep over again, as DD plays up to her in particular.
Good luck. It's no fun at all.