I originally meant to post this in this thread but w/e.
Alpha has always struck me as sort of material-driven quadra. There was a group of kids in my class who were always impeccably dressed, who had mild culturally-appropriate humour and who were in the loop with all the new movies and all the new music and who looked so clean and perfect all the time -- those were the Alpha kids. I think every school has a clique like this. Of course every quadra has their own definition of fun so maybe calling it the "fun" quadra isn't right and I think a better term would be "light-hearted". Alpha is the light-hearted quadra, the people who like to keep things light and airy. There's this weird sort of consistency in how they approach situations/interests which is to pursue them as long as they fit the overall aesthetic appeal of their personality and/or group to stay within the confines of what they identify with, which could range from being an otaku to being a goth and everything in-between. I don't see Alphas as particularly weird but I see a propensity towards being interesting, like they try hard to come off as "interesting people" by dressing perfectly and having streamlined interests and being generally filtered. (Note: This is slightly biased towards Alpha SFs.)
Beta is simultaneously very similar to and very different from Alpha. There was another group in my class who would sit in the back of the class, make noises or pass comments and do other "funny" things. They would disrupt everyone's concentration and piss teachers off and really just make it annoying as hell for everyone who was not in on the joke. They were also always dressed well but it seemed less ubiquitous than Alphas, looked less like actors-having-walked-out-of-a-movie-set and more like people who knew how to impress. They were the Beta kids. The weird thing about them was that they didn't do all that because they had anything to rebel against -- they did it because they thought it's just what people do, just what they had to do. If they didn't do it, no one else would. If they saw an opening to slip their humour into, they would just do it, because once the moment is over you might regret never doing it. In the betas I've met, there's always this disinclination to take part in "group activities", the kind of stuff that is orchestrated solely for the purpose of "having fun". To them it seems fake and put on, and they much rather prefer to be swept in by the moment, to live the present to the fullest intensity, to do everything, to feel everything and leave nothing to regret. (Note: Slightly biased towards Beta EXXx, but similar themes run course in both Beta IXXx)
I don't think there was a Gamma equivalent in my school, lol. I mean there were a few people who were distinctly Gamma but I don't think they ever coalesced into a clique or anything. I remember there were these two guys, Se-ESFp and Te-INTp, who would sit at the far end of the class who wouldn't talk very much but when asked to parody someone, they would quickly jump up and do hilariously well. In strict contrast with Betas who relied, I suppose, on subtler cues to join in on the fun where they saw an opening, the Gammas would wait for the "right time" to be funny, to make noises or joke around, when they could be absolutely sure they wouldn't annoy anyone. None of the Gammas I knew cared much for clothing or whether they looked "good", they dressed kind of shabby except when they were formally required to dress well, there was no "aesthetic tone" to the things they did or said and neither did they put much value in those things. Gammas struck me as the most "be real" quadra, as the people least judgmental of differences or disabilities or things like that.
The Deltas I was around in school were generally non-disruptive, didn't make much noise not because they were scared of getting caught but because they saw it as an agreement between themselves and the teacher to not disturb the class (I see an animosity towards such "made up" agreements in Betas, similar to how they don't appreciate any "made up" attempts to increase affinity between people, whereas Deltas process things in terms of these agreements). They weren't entirely dull, though, they were appreciative of novelty and weirdness (as long as it didn't break any of the aforementioned agreements) and despite being so "earthy", they had secret fetishes for fantasy fiction or prog rock or science fiction or so. One thing I find about deltas is that there's a ubiquitous conflict between what they embrace and what they reject; almost all of them simultaneously reject social customs/norms and follow them at the same time; they long to get away from social contracts, having to dress well for other people, having to be nice when they don't want to, and yet still do all of it. If Gamma's primary motif is "be yourself", then delta's would be "be as good as your word".
I think I described other quadras much better than my own. Oh well.