Petri nets have proven effective in modeling discrete dynamic systems whose state evolves over configuration space in discrete time steps. In this paper, we construct a Petri net that models the flow of cars on a roundabout with n entrances, symmetric about its centre, and analyze the semigroup related to the constructed Petri net. The paper focuses on investigating the complexity of the model in Krohn–Rhodes terms using the holonomy decomposition of the roundabout transformation semigroup and the general properties of such Petri nets for different n. Analysis of some components in the Petri net has given us insights into their role in defining the complexity of the model. We study the natural subsystems of the state space and the permutator groups acting on them, some of which are non-abelian or non-solvable. We catalog these dynamical patterns of changes in the number and location of vehicles on the traffic roundabout described by permutator groups.