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Flimnap is the treasurer of Lilliput; he quickly takes against Gulliver thanks to some totally absurd rumors floating around court that Gulliver is having an affair with Flimnap's wife. Seriously, no – she's six inches tall to his six feet. Anyway, Flimnap joins the plots of Skyresh Bolgolam, admiral of Lilliput and also irrationally set against Gulliver.
Like the Emperor, Flimnap and Skyresh Bolgolam both represent actual living people of Swift's period. Flimnap is Robert Walpole, the Whig leader who fell out of power in 1717 only to be saved by George I's mistress, who helped him back into power in 1721. Thus, the King's pillow, which breaks Flimnap's fall during his rope dance, is probably a swipe at Walpole's dubiously engineered return to office through the King's mistress (citation: Robert Greenberg, Editor, Gulliver's Travel. New York: Norton, 1961, 22).
As for Skyresh Bolgolam, he could be the Earl of Nottingham, the peer who opposed Swift's candidacy for bishop. He might also be the Duke of Marlborough, a major contributor to England's part in the War of the Spanish Succession (source). Either way, Bolgolam is a bad cookie. He has no particular reason to hate Gulliver, but he does, from the first minute he sees him. Bolgolam's prejudices lead directly to Gulliver's death sentence from the Emperor.