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The phrase "a third division of" is correct and usable in written English.
It can be used when referring to a specific segment or category within a larger classification or organization.
Example: "The organization has a third division of volunteers dedicated to community outreach programs."
Alternatives: "a tertiary division of" or "a third category of".
Exact(3)
These were succeeded by a third division, of picked men, native Macedonians, the choicest for courage and strength, in the prime of life, gleaming with gilt armour and scarlet coats.
The Americans would keep a division in and around Baghdad Britainn would command a multinational division in the south near Basra and Polandd would command a third division of troops from a variety of nations.
The Americans would keep an Army division in and around Baghdad, and likely some presence in northern Iraq Britainn would command a multinational division in the south near Basra and Polandd would command a third division of troops from a variety of nations.
Similar(57)
A first division of the city can be done considering areas to be inside or outside the GRA (a circular freeway of approximately 68 km of length).
But Green warned the SPL not to assume that SFL clubs would want to join their ranks should they pursue a second division of their own.
The new platform, also known as X.com, will be a fourth division of the company after eBay Marketplaces, PayPal, and GSI Commerce.
Database searches identified proteins with similarity to AtMHX in a fourth division of the plant kingdom, namely Chlorophyta (green algae).
"If the refereeing had been of a standard even close to the third division of a European league, Clint Dempsey would have been sent off after 20 seconds and the US would have been playing with 10 men for the game, so let's not think that EVERY crap decision went against the US in that match eh?" wibbles Gary Walker.
In March last year Ziya Eren, a Turkish steel magnate, completed a takeover with ambitions to raise Crawley to the level of Turkey's Kayseri Erciyesspor, a second division club of which he is president.
The world's 221 international airlines are coalescing into a first division, consisting of four groups (see chart), and a long tail in the second division, keen to win promotion or confused about what to do next.
In December 1912, the Ottoman fleet was reorganized into an armored division, which included Barbaros Hayreddin as flagship, two destroyer divisions, and a fourth division composed of warships intended for independent operations.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com