# Monster group

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## Definition

This group, termed the Monster or Monster group (and denoted by the shorthand $M$), is the largest sporadic simple group. It has order: $808017424794512875886459904961710757005754368000000000 = 2^{46} \cdot 3^{20} \cdot 5^9 \cdot 7^6 \cdot 11^2 \cdot 13^3 \cdot 17 \cdot 19 \cdot 23 \cdot 29 \cdot 31 \cdot 41 \cdot 47 \cdot 59 \cdot 71$.

The prime divisors of the order of the monster group are precisely the same as the supersingular primes.

The monster group is a sporadic simple group, and is the sporadic simple group of largest order. A sporadic simple group that is not isomorphic to a subgroup of the monster group is termed a pariah.

## Arithmetic functions

### Basic arithmetic functions

Function Value Similar groups Explanation for function value
order (number of elements, equivalently, cardinality or size of underlying set) 808017424794512875886459904961710757005754368000000000 groups with same order

### Arithmetic functions of a counting nature

Function Value Similar groups Explanation for function value
number of conjugacy classes 194 groups with same order and number of conjugacy classes | groups with same number of conjugacy classes See element structure of monster group, linear representation theory of monster group

## GAP implementation

This group is too large to be stored and actively manipulated in GAP. However, some information about the group is stored in GAP under the symbol "M" -- for more, see linear representation theory of monster group.