Archie likes to believe that he is in command of the Vigils in The Chocolate War. Carter is the group's president, but Archie thinks of him as as figurehead with no real authority. Carter is respected around the school for his athletic prowess and boxing skill, but Archie thinks of him as a sort of brute.
In Chapter 27 Archie disapproves of Carter's readiness to beat up an insolent junior, Frankie Rollo.
When Archie fails to coerce and intimidate a student brought in to a meeting of the Vigils and that student begins to disrespect the group, Carter steps in and punches the student in the stomach.
Archie is humiliated for two reasons by this episode. First, he failed to control the situation with his intellect and so lost status in the eyes of the group. This had never happened before. Archie had been able to talk his way around any problems and get his way. This time, he failed.
In losing some measure of his status, Archie also lost control of the situation. Carter resorted to violence, which is against Archie's code, and Carter took control of the group, effectively taking over Archie's role as the leader.
Carter warns Archie that he'll be through in his position with the Vigils if he does not find a way to make the chocolate sale work and also solve the problem that Jerry has created for the group. This warning is also humiliating for Archie, but he soon finds a way to regain the respect and control of the Vigils.