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The one behind the real assassination were Imai Nobuo and several men from the Mimawarigumi, under orders from their commander, Sasaki Tadasaburo.
Nobuo confessed later that he was the one behind the assassination, and later got arrested and jailed.
The Shinsengumi itself
Based on the eponymous unit, formed in 1863 by the Tokugawa shogunate, in response to the murderous acts by the loyalists to Japan's Emperor. Initially named the Roshigumi, and were a group of 234 ronin, under Matsudaira Tadatoshi's nominal command.
Made a 6x6 of my personal favorite female anime/manga characters (in order).
Was gonna make it a top 25, but felt it wouldn't enough...
From Gintama vol. 64, Q.C. 216, Sorachi explained Takasugi's poem. Here it is, in case you haven't read it yet.
However, the one who got accused of the assassination was Kondo Isami, who ended up getting executioned in his place.
The events from Farewell Shinsengumi arc were partly inspired by this event. Sorachi reversed the events, having Isaburo kick the bucket in Kondo's place.