In a manner that opposes or acts against government, governmental authority, or state power.
From anti- + governmental + -ly (adverb suffix). Built by adding -ly to antigovernmental to create an adverb describing how something is done in opposition to government.
English lets us stack affixes indefinitely—antigovernmentally packs anti-, government, -al, and -ly into one word, showing how productive English is at creating precise shades of meaning!
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