(nice pun, eh?)
well I just realized that all my life I've had groups of friends that were in different quadras. With the exception of two IEIs that are lifelong friends (but those friendships have always been one-on-one, not part of any group), I've never had a group of beta friends.
I'll bet it would be fun.
Anyway, I think that when you grow up without being a part of a group of your own quadra, you start to feel like you're... a bit weird or out of place somehow. Being a 4, I'm prone to seeing myself as different or special anyway but I think that feeling might be magnified if the things you value aren't affirmed by others consistently. When I think back on my groups of friends, it was mostly gammas. They were great people (esp. the ISFjs) but it explains why I so often felt like a fish out of water.