# Monster group

From Groupprops

This article is about a particular group, i.e., a group unique upto isomorphism. View specific information (such as linear representation theory, subgroup structure) about this groupView a complete list of particular groups (this is a very huge list!)[SHOW MORE]

## Contents

## Definition

This group, termed the **Monster** or **Monster group** (and denoted by the shorthand ), is the largest sporadic simple group. It has order:

.

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.