Monster group

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VIEW DEFINITIONS USING THIS AS EXAMPLE: All terms |Group properties |Subgroup properties|
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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

Linear representation theory

Further information: 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.

External links

• M on the Atlas of Finite Groups