---
tag: C9VY20TAPT
kind: object
name: Monster group
msc: [20D08, 11F22]
uses: [PZ49Z4CBFZ, 0YMYMNQ0QZ, 0B85S2V2DR, ZMPYZ8KD9N, FX3NEFYFFN, FEC2XH822R]
sources:
  - url: https://en.wikipedia.org/wiki/Monster_group
same_as:
  wikidata: Q392663
  wikipedia: Monster group
---

## Description

The Monster group $\mathbb{M}$ is the largest of the $26$ [[sporadic groups|PZ49Z4CBFZ]].
It is a finite [[simple group|0YMYMNQ0QZ]] of [[order|0B85S2V2DR]]
$$\abs{\mathbb{M}} = 2^{46} \cdot 3^{20} \cdot 5^9 \cdot 7^6 \cdot 11^2 \cdot 13^3 \cdot 17 \cdot 19 \cdot 23 \cdot 29 \cdot 31 \cdot 41 \cdot 47 \cdot 59 \cdot 71,$$
which is $808017424794512875886459904961710757005754368000000000$, about $8 \cdot 10^{53}$.

## Remarks

The smallest nontrivial [[irreducible representation|FX3NEFYFFN]] of $\mathbb{M}$ over $\C$ has dimension $196883$.
Robert Griess constructed $\mathbb{M}$ as the automorphism group of a commutative non-associative algebra of dimension $196884$, the [[Griess algebra|ZMPYZ8KD9N]].
Its existence had been predicted earlier by Bernd Fischer and by Griess.
The [[$j$-invariant|FEC2XH822R]] has the expansion $j(\tau) = q^{-1} + 744 + 196884 q + \cdots$ with $q = e^{2 \pi i \tau}$, and $196884 = 196883 + 1$.
This coincidence led to the [[monstrous moonshine|E0Y7Y75T6C]] conjecture of Conway and Norton, proved by Richard Borcherds.
Twenty of the $26$ sporadic groups are subquotients of $\mathbb{M}$, that is, quotients of subgroups of $\mathbb{M}$; the other six are the [[pariah groups|ZYZ5FYG7QC]].

## Explanations

### L1 human

The Monster is the largest of the exceptional finite simple groups, with about $8 \cdot 10^{53}$ elements.
Its smallest faithful action by complex matrices needs matrices of size $196883 \times 196883$.
It contains most of the other exceptional simple groups, and it is linked in surprising ways to number theory through monstrous moonshine.