\(G_2\) is the symmetry group of a nonassociative algebra called the 'octonions'. But it also appears as the symmetries of a ball rolling on another ball!
There is a geometry where the 'points' are ways one ball can touch another fixed ball, and the 'lines' are the ways it can roll along a geodesic without slipping or twisting.
If we set things up just right, this will be a \(G_2\) geometry!