Carter was a student at the Experiment House school, where Eustace also attended. He was also a member of The Gang; a group of students there who used to bully their fellow pupils, including Eustace Scrubb and Jill Pole.
He was never punished by the Head-teacher for his bullying and disgraceful behaviour because the Head believed that children should be allowed to do what they wanted, and thought he and his gang were 'interesting' psychological cases.
Apparently, he was also cruel towards animals, as he presumably tormented a rabbit, which Eustace stood up to him about.
Carter and his gang were presumably later expelled from Experiment House.