From pages 62 and 64 from Book 3: Extermination.
Imperial Fist observed strategic tendencies: ship-bourne assaults and boarding actions, defensive and fortification operations in extremis, stronghold assaults, void warfare.
The fists have a lot of assault units and special weapon teams dedicated to killing vehicles and fortifications.
After those, there are a lot of heavy weapons - on vehicles and carried by marines, but missile launchers and lascannons rather than autocannons and multi-melta.
They are the strongest proponents of tactical dreadnought armour (they rocked the indomitus pattern in large numbers early on) and at the time of the betrayal, they were the only legion to have Abstina and Iliastus pattern assault cannons on their terminators
A company could vary in size from 'a few squads' to 'several hundred' marines.
The size changes according to recruitment and casualties.
Each company is a battalion regardless of actual size and two battalions make a regiment or 'crusade' or 'household' - larger companies were formed as needed.
There is a temple on board the 'Phalanx' which is dedicated to renewal of oaths of allegiance to the great crusade/emprah and only Dorn, omnissiah and Templar brethren could go without a direct invitation.
The first company rarely drop below 1000 and are drawn from all over the IF legion - bearing in mind that each company tends to get made from recruits from a single conflict, that first company is going to be a mess of different traditions, colours, languages and combat preferences.
There is a direct link from Dorn through the 'senior captains' - no fixed advisors or senior lieutenants like other legions.
(there is actually a second layer between Dorn and the captains made up of fleet masters, siege masters, theatre commanders, marshals and the 'first captain' - they take command of a combat zone rather than individual units)
Under the 'senior captains' are the 'line captains' and 'centurions' who actually command the battalions and regiments.
The IF generated impromptu titles for marines taking command of permanent fortresses that they built.
The last two titles they used are 'Castellan' and 'Legion Seneschal' and these titles were retained by the honoured marine regardless of posting/mission/command.
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