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I'm creating a simplified version of creating 4 groups of 6 from 24, using 6 pairings such that every letter is matched with every other at least once but not twice. There has to be a good method to this.

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So your ultimate goal is the 24 team schedule, divided into groups of 6?

Unfortunately, the area of math that you are interested in (Combinatorial Design Theory) is more devoted to finding solutions that work out perfectly, i.e., when each team plays the others exactly once. For dividing into groups of 3, this is known as the Kirkman school girl problem and has a perfect solution if and only if you have 6n+3 teams. I'm assuming that you want each team to play at least once but no more than twice in your 24 team model because it is trivial to show that it isn't possible to have an exact solution, but that also means that the literature has tended to skip your problem.

I would be tempted to look at individual movements for 24 players at bridge and see if you get what you want from there. Have all the people playing N on board 1 be 1 group of 6, the people playing E on board 1 a 2nd group, etc. I don't know if this will work or not. A couple of different 24-player movements are at this webpage