government office
衙 centers on an official administrative place: a government office or bureau where public business is conducted.
衙 combines 行 (to go, movement) with 吾, which likely contributes the sound. The compound originally referred to a place where officials gather or move about, hence a government office.
The outer 行 suggests a building or enclosure, and inside 吾 (which looks like a person with a mouth) can be seen as an official speaking. Picture a government office where officials walk around and give orders.
For ガ, imagine a government office guarded by a stern guard: guard -> ガ, and the guard stands at the entrance of the 衙.
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