In this work, we aim to investigate the mechanism of a multi-group epidemic model taking into account the influences of logistic growth and delay time distribution. Despite the importance of the logistic growth effect in such models, its consideration remains rare. We show that ${\mathcal{R}}_0$ has a crusher role in the global stability of a disease-free and endemic equilibria. That is, if ${\mathcal{R}}_0$ is less than or equal to one, then the disease-free equilibrium is globally asymptotically stable, whereas, if ${\mathcal{R}}_0$ is greater than one, then a unique endemic equilibrium exists and is globally asymptotically stable. In addition, we construct suitable Lyapunov functions to investigate the global stability of disease-free and endemic equilibria. Finally, we introduce numerical simulations of the model.