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When does the penitential rite take place in a Catholic mass?
In the beginning, call the Introductory Rite: after the initial greeting, and before "Gloria" hymn (if it is prescribed) and the Collect (Opening prayer).
Before I answer your question, I think it would be helpful to explain what the "penitential rite" is. The penitential rite is part of the Introductory Rites of the Mass that come before the Liturgy...
There is no "Roman Catholic Rite". There is a Latin Rite, Byzantine, Armenian, Chaldean, Coptic, Ethiopic, Malabar, Maronite, and Syrian Rites. They are all Catholic Rites.
The central action that takes place during the Mass is the re-presentation of the Sacrifice of Our Blessed Lord on Calvary and the application of the graces therefrom to the faithful who are present....