Among all finite ‘symmetry atoms’ — technically known as ‘finite simple groups’ — are 26 exceptions, called sporadic groups. The largest is The Monster, which contains all but six of the others, though each one was discovered independently, and the story of these discoveries is described in my book Symmetry and the Monster.
The size of the Monster is 246.3126.96.36.199.188.8.131.52.184.108.40.206.59.71, which works out to be 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000.
Being the largest exception was its first claim to fame, but the Monster soon showed connections to number theory, and to string theory in mathematical physics. These Moonshine connections were entirely unsuspected when the Monster first emerged, via one of its cross-sections later called the Baby Monster. The Baby needs more than 4,000 dimensions in which to operate, but the Monster itself needs 196,883, a number that is the product of the three largest prime numbers that are divisors of its size, namely 47, 59 and 71.
The existence of the Monster was proposed by Bernd Fischer, immediately after he discovered the Baby Monster, and it was later constructed by Robert Griess as the symmetry group of an algebra structure in 196,884 dimensions. His work split the space into three subspaces, and his main task was to show there were symmetries intermingling these subspaces. The dimensions of the subspaces are:
98,304 + 300 + 98,280 = 196,884
The first number 98,304 = 212 × 24 comes from the Golay code in 24 dimensions.
The second number 300 = 24 + 23 + 22 + … + 3 + 2 + 1 is the dimension of the space of 24-by-24 symmetric matrices.
The third number 96,280 = 196,560 ÷ 2 comes from the Leech Lattice in 24 dimensions, where there are 196,560 vertices closest to a given vertex, forming 98,280 diametrically opposite pairs.
The Monster’s Character Table
The finite symmetry atoms are very large, and data about each one is encoded into a character table — a square array of numbers, rather like a giant sudoku puzzle. The Monster’s table has 194 rows and columns, and the Moonshine connections showed that the first column generates an important sequence of numbers in number theory (the coefficients of the j-function). Other columns can be used in a similar way, and these moonshine connections eventually created a link to the mathematical physics of string theory.
There are still mysteries associated with the Monster. Here is one. The 194 columns of the Monster’s character table span a space of 163 dimensions. The number 163 is well-known in number theory because the square root of -163 yields an extension of the rational numbers having unique factorisation, and 163 is by far the largest integer having this property.