When I was an undergrad GSS, we did not belong to any "Greek council". There was only us and APO, and we answered to the Student Activities/Leadership Development council. I would guess that a lot has to do with campus regulations. But GSS is not part of any national umbrella organization.
We assisted the Lambda Theta Phi guys during their pre-colonolization process during my last year on campus, but GSS disbanded the semester after I left, so I don't know what the story is today, although I know several other Latin and MC groups, and an NPC colonized since I graduated.
I have to ask, and I mean this in the most unassuming way - why so much interest in Gamma Sig?
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