Cheaters Gentleman’s Club in Providence was shut down Monday after an underage girl was found dancing there and after incidences of prostitution were discovered, officials said. The Allens Avenue strip club had its liquor and adult entertainment licenses suspended Monday and must remain closed for 45 days, said Andrew Annaldo, chairman of the city’s Board of Licenses. Cheaters was also given the maximum fine of $5,000, Annaldo said. Over the summer, it was discovered that the club had hired a 15-year-old girl as a dancer. There had also been one or two instances of prostitution, unrelated to the teenage employee, Annaldo said.