The incident took place yesterday when she had gone to take water from a hand pump near her house, they said.
The girl was taken to a house by the youths and raped at gun point. One of the perpetrators made a video of the incident, Circle Officer Rajiv Kumar Singh said.
He said a case was registered against the three youths on a complaint by the girl's father. The accused are at large, Mr Singh said.
They have been booked under relevant sections of the Indian Penal Code (IPC), including rape, and provisions of the Protection of Children from Sexual Offences (POCSO) Act, the officer said.
Police are investigating the matter.
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