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On the 13th Floor of Police Headquarters in downtown Manhattan, there is an ink drawing of a stern-looking man with a piercing gaze.

It is Jacob Hays, and his image is the first in a long series of framed portraits of the department’s police chiefs.

For four decades before the Police Department was formed in 1845, Mr. Hays served as New York City’s highest-ranking law enforcement official: a now-defunct position known as high constable — a “top cop” post that also required street-level patrol and detective work.

“You could probably more accurately describe him as the whole Police Department,” Edward Conlon, a communications director for the department, said. “He was doing the job that cops and detectives do.”