I 夷 is a derogatory term for 'foreigner' or 'barbarian'
Combined the phrase literally means 'expel-the-foreigners'
Shishi 志士 means 'patriot', though in the Gintama context 'rebel' may be more appropriate
There are actually a number of different factions within the Jouishishi rebellion, some more extremist than others. Katsura's faction grew moderate as the anime progressed, from the the initial opening scene in Episode 5 where Katsura used terrorist-like tactics like time bombs liberally, to the Benizakura Chapter Episode 61 where he pleaded with Takasugi Shinsuke that there must be other ways of changing the world other than destroying it all. However, he was still happy to deploy bombs liberally against Shinsengumi 真選組.
Their anti-government and/or terrorist activities made them the arch nemesis of the Shinsengumi 真選組, who had made it their mission to wipe out the rebellion movement. However, the extremist factions are known to have secret dealings with the Bakufu government, for example in Episodes ?.verification/corroboration needed
Members
Katsura Faction 桂一派
Moderate Factions
Within Katsura's faction, as part of his recruitment policy he conducts "Jouishishi Employment Examination" 攘夷志士採用試験 for candidates who wish to join his faction. The examinations comprise face-to-face interview, written examination and for those who pass the afore-mentioned 2 parts, they can proceed to the practical examination, conducted by Jouishishi's in-house trainer called Dragon Leader, assisted by Debugon and Katsura Kotarou himself. The entire examination sequence was showcased in Episode 109.
Extremist Factions
Kiheitai 鬼兵隊 literally means "Army of Demon Soldiers". Resurrected from the 義勇軍 gi yuu gun (volunteer army) from the Joui Wars, this is an extremist group that stops at nothing in its mission to 'destroy the world', quoting the leader Takasugi Shinsuke 高杉 晋助.
Extremist Factions
Apparently they hold a similiar amount of influence like Katsura. After Katsura and Gintoki destroyed the Tenseikyou drugs and defeated the Harusame, Kaitendou knew Kinya 禽夜 was involved and they planned to assassinate him. The Shinsengumi protectd Kinya even though they hated him. The plan foiled as Kondo covered a bullet for Kinya and the Shinsegumi defeated Kaitendou.
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Justaway Factory
Extremist Factions
Banzou was building up an army of Joui Patriots by recruiting former Jouishishi, he was planning on overthrowing the Bafuku. He was secretly developing weapons and Justaway. He recruited Sakata Gintoki and Kondo Isao which turned out the two were mostly responsible for destroying their operation.
Extremist Factions
The original leader was Tendou Koda until he was killed by the Shinsengumi in the Rokkaku Inn incident. They originally planned to set fires around Edo as a distraction so they can assassinate the Shogun. The first devision led by Okita Sougo charged in Rokkaku Inn, there was 36 catastrophic death between the Soukai and Shinsengumi which only Tendou Soutatsu, Okita Sougo and Kamiyama survived. Tendou Soutatsu would become the next leader and plans to kill Sougo for revenge by using Rokkaku Kirie.
Extremist Factions
Tanaka Kohei also known as Kohei the Manslayer is the leader of the Akatsuki faction. He was surrounded by 40 Bafuku Officials and was able to kill 7 before being captured.
Tanaka Kohei
Blues Crew
Extremist Factions
They tried to take over Great Edo Hills building along with Matsudaira Kuriko and others as a hostage.
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Check It Out
Extremist Factions
They used to be a thug gang that disses the Bafuku with their rap but after continued strife and expansion, they transformed into a group of radical rebels.