G2 shall be my favorite group! Well, for now. For a while. Or maybe for longer. It's the smallest of the exceptional Lie groups. Found 1893 by Élie Cartan. It was he who suggested to think about it in terms of rolling balls. https://en.wikipedia.org/wiki/G2_%28mathematics%29
Picture two spheres, one three times larger than the other. Imagine them rolling on another without slipping nor twisting. Rolling surfaces have their own branch of mathematics: contact geometry!
Also funny: since the big ball isn't really a ball, but a projective plane, that that also played a role in my construction of SO(3) above. No rotation group corresponds to RP² it somehow seems to sit between SO(2) and SO(3)!