C172 instructor pilot with trainee reported a lose of engine power in their single engine aircraft. The instructor; with assistance from ATC; divert to a closer airport and landed safely.

Narrative

I am writing to formally report an incident that occurred during an instrument flight lesson with the tail number Aircraft X. I am providing this statement in order to document the details of the incident. At approximately XA30z; both the student and I experienced a sudden loud noise followed immediately by severe shaking of the aircraft. The incident took place when we were flying at 4000 ft. indicated; on a direct course from ZZZ to ZZZ1; with a heading approximately 287 degrees magnetic. Upon sensing a problem; I promptly assumed control of the aircraft from the student. I would like to emphasize that there were no indications or signs of engine malfunction prior to the occurrence. We meticulously followed all pre-departure procedures; including run-up and before takeoff checks; in accordance with the on-board checklists and the schools policies and procedures. As I assumed control; my immediate objective was to descend safely to VFR or VMC and notify ATC of my intentions. I promptly informed ATC of the situation on frequency XXX.XX. Despite running the rough running checklists; we were unable to resolve the problem. ATC guided me to ZZZ2; which was the closest airport to our location. At that point; exercising my authority as the pilot in command; I [requested priority handling] and set my transponder code to 7700. Upon visual contact with the airport; I made the decision to maintain altitude as best as possible and conduct an overflight to assess the field conditions. Once satisfied with the condition of the runway; I maneuvered to land on the uphill runway. The landing was executed without significant difficulty; and we vacated the runway promptly and safely. No further actions were taken following the landing. I trust that this formal report provides a concise and accurate account of the incident that transpired during the aforementioned flight. Please do not hesitate to contact me if any additional information or documentation is required. Thank you for your attention to this matter.