"Division pistol" is a title based upon conjecture.
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Some humanoid operatives of the Division carried laser pistols. Although such laser pistols, and similar laser rifles, appeared to be standard Divison issue, they were not compulsary and some agents - such as Karvanista - chose to use their own weapons.
A member of the Fugitive Doctor's team carried a laser pistol during the Siege of Atropos. (TV: Once, Upon Time)