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The finite simple groups have been completely classified; there are several infinite families of finite simple groups, plus 26 "sporadic groups" that don't follow any apparent pattern. The Monster group is the largest of these sporadic groups. See classification of finite simple groups.
The Monster was found by B. Fischer and R. Griess in 1973. It can be constructed as a group of rotations in a space of dimension 196,883 over the rational numbers.
The Monster group prominently features in the Monstrous Moonshine conjecture which relates discrete and non-discrete mathematics and was proven by Richard Borcherds in 1989.