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The Gestapo was the state police and intelligence agency of Germany during the Nazi era of 1933 to 1945. They were considered one of the most feared and ruthless such organizations in Earth history.
James Kirk questioned Melody Sawyer's decision to follow the orders of her AreoNav superiors unquestioningly to the behavior of the Gestapo, as well as to the troops serving under Colonel Green. (TOS novel: Strangers from the Sky)
Federation historian John Gill created a Gestapo, along with a National Socialist government, on planet Ekos in the mid 2260s. (TOS episode: "Patterns of Force")
Gestapo officers played a role in a USS Voyager holodeck program set in France, 1944. (VOY episode: "The Killing Game")
During his accidental visit to the "mirror universe," Commander Montgomery Scott compared security aboard the ISS Enterprise, under the command of Hikaru Sulu, to the Gestapo. (TOS episode: "Mirror, Mirror")
In another alternate timeline, Gestapo officers searched the streets of 1944 Brooklyn for Captain Jonathan Archer, going house-to-house and shooting people at random. (ENT episode: "Storm Front")