Officials with the organization had no immediate comment on the crash, the Associated Press said.
The foundation's Facebook page said the group was traveling to Harvard to "visit the campus, sit with the office of cultural advancement, followed by a tour of the campus . . . followed by Harvard Square (shopping, eating, site seeing . . . etc. . . .)"
The crash occurred shortly before 8 p.m. Saturday when the bus struck an overpass in the Allston neighborhood.
Massachusetts State Police said the bus crashed into the Western Avenue Bridge on the eastbound side of Soldiers Field Road, by the Charles River.
Authorities said the Calvary Coach bus did not belong on that road, which was not authorized for oversized vehicles, and state police were investigating whether charges were warranted against the driver, who apparently failed to see a sign prohibiting the vehicle from taking that route.
Ray Talmedge, owner of the Philadelphia-based Calvary Coach Bus, said his driver was being interviewed by police. Talmedge said the driver, who also drives a school bus, didn't know about the road restrictions.
Of the 34 passengers injured, state police said, three had serious injuries. State police did not know late Saturday night whether the seriously injured were students or adults.
The Boston Fire Department said some passengers were trapped immediately after the accident. Photos on the department's website showed firefighters lifting victims strapped to boards through the roof of the bus, which bears a Calvary Coach logo.
Four people were taken to Brigham and Women's Hospital for treatment, and 19 were taken to Beth Israel Deaconess, according to a Brigham and Women's Hospital spokeswoman. Boston EMS said the victim with life-threatening injuries was rushed to Boston Medical Center. Another victim was said to have been transported to Massachusetts General Hospital.
Contact Miriam Hill
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Associated Press and the Boston Globe contributed to this article.