A Fabulous Biplane
Just read this paper on the (11,5,2)-biplane. A biplane is like a projective plane except that each pair of points is contained in precisely two lines and each pair of lines intersects at exactly two points. It's an exceptional object in the sense that a whole slew of exceptional objects can be constructed from it: the S(5,6,12) and S(5,8,24) Steiner systems, the binary and ternary Golay codes and the Mathieu groups.
I know it's all kid's stuff but I never studied (finite) groups beyond a first course and so little things please me!