Their rivalry is dated back to kindergarten, when GiGi spilt juice on Alex's mat and then told everyone Alex had an accident. She also targets Harper in the episode Alex's Choice, and invites Harper to her annual tea party to crown her the Biggest Loser, showing she doesn't see Harper as a challenge. Harper also states that "GiGi backwards is still GiGi", showing it must be the letters.
GiGi does not torment Alex and Harper alone; she has a duo of girls that bow down to GiGi's many whims. GiGi hangs out with two girls who Alex and Harper refer to as the "Wannabes." The girls dress similarly to Gigi and underwent plastic surgery on their noses to look more like her. They will do anything she wants, and when GiGi's not there Alex and Harper tell them that GiGi wants them to shave their heads, which they do to GiGi's horror and also put there shoes on their hands and put on yarn wigs and the wannabees were smart enough to know GiGi hates yarn and almost knows that Alex and Harper were playing revenge with them.
GiGi's quest to make Alex look bad almost outs the Russos as wizards when she gets into Alex's private journal and lives in Alex's medieval fantasies. It is then that she discovers Alex's crush on Dean Moriarty. Other than Harper and Zeke Beakerman, she can be considered the character closest to finding out the Russo secret.
It is also implied that GiGi is wealthy, as mentioned by one of the girls in Alex's Choice, where she said she "If my parents could afford the same dermatologist as GiGi's, I'd be way prettier than her." Typical episodes involving GiGi, depict her trying to humiliate Alex and/or Harper, and receiving poetic justice, usually from Alex's end. It's also made clear that GiGi is the most popular girl at Tribeca Prep.