Editing Monster group (section)

==Representations==
The minimal degree of a [[Faithful representation|faithful]] complex representation is 47 × 59 × 71 = 196,883, hence is the product of the three largest [[prime divisor]]s of the order of M.
The smallest faithful linear representation over any field has dimension 196,882 over the field with two elements, only one less than the dimension of the smallest faithful complex representation.

The smallest faithful permutation representation of the monster is on

:&nbsp;&nbsp;&nbsp;97,239,461,142,009,186,000
: = 2<sup>4</sup>{{·}}3<sup>7</sup>{{·}}5<sup>3</sup>{{·}}7<sup>4</sup>{{·}}11{{·}}13<sup>2</sup>{{·}}29{{·}}41{{·}}59{{·}}71 ≈ 10<sup>20</sup>

points.

The monster can be realized as a [[Galois group]] over the [[rational number]]s,{{sfn|Thompson|1984|p=443}} and as a [[Hurwitz group]].{{sfn|Wilson|2001|pp=367–374}}

The monster is unusual among simple groups in that there is no known easy way to represent its elements. This is not due so much to its size as to the absence of "small" representations. For example, the simple groups A<sub>100</sub> and SL<sub>20</sub>(2) are far larger but easy to calculate with as they have "small" permutation or linear representations. [[Alternating group]]s, such as A<sub>100</sub>, have permutation representations that are "small" compared to the size of the group, and all finite simple groups of [[group of Lie type|Lie type]], such as SL<sub>20</sub>(2), have linear representations that are "small" compared to the size of the group. All sporadic groups other than the monster also have linear representations small enough that they are easy to work with on a computer (the next hardest case after the monster is the baby monster, with a representation of dimension 4370).

=== Computer construction ===
Martin Seysen (2022) has implemented a fast [[Python (programming language)|Python]] package named [https://mmgroup.readthedocs.io/ mmgroup], which claims to be the first implementation of the monster group where arbitrary operations can effectively be performed. The documentation states that multiplication of group elements takes less than 40 milliseconds on a typical modern PC, which is five orders of magnitude faster than estimated by [[Robert Arnott Wilson|Robert A. Wilson]] in 2013.<ref>{{cite web |url=https://mmgroup.readthedocs.io/en/latest/api.html |title=The mmgroup API reference |last=Seysen |first=Martin |access-date=31 July 2022}}</ref><ref>{{cite arXiv |last=Seysen |first=Martin |author-link= |eprint=2203.04223 |title=A fast implementation of the Monster group |class=math.GR |date=8 Mar 2022}}</ref><ref>{{cite arXiv |last=Seysen |first=Martin |author-link= |eprint=2002.10921 |title=A computer-friendly construction of the monster |class=math.GR |date=13 May 2020}}</ref><ref>{{cite arXiv |last=Wilson |first=Robert A. |author-link=Robert A. Wilson (mathematician)|eprint=1310.5016 |title=The Monster and black-box groups |class=math.GR |date=18 Oct 2013}}</ref> The mmgroup software package has been used to find two new maximal subgroups of the monster group.{{sfn|Dietrich|Lee|Popiel|2025|}}

Previously, Robert A. Wilson had found explicitly (with the aid of a computer) two invertible 196,882 by 196,882 matrices (with elements in [[GF(2)|the field of order 2]]) which together [[Generating set of a group|generate]] the monster group by matrix multiplication; this is one dimension lower than the 196,883-dimensional representation in characteristic 0. Performing calculations with these matrices was possible but is too expensive in terms of time and storage space to be useful, as each such matrix occupies over four and a half gigabytes.{{sfn|Borcherds|2002|p=1076}}

Wilson asserts that the best description of the monster is to say, "It is the [[automorphism group]] of the [[monster vertex algebra]]".  This is not much help however, because nobody has found a "really simple and natural construction of the monster vertex algebra".{{sfn|Borcherds|2002|p=1077}}

Wilson with collaborators found a method of performing calculations with the monster that was considerably faster, although now superseded by Seysen's abovementioned work. Let ''V'' be a 196,882 dimensional vector space over the field with 2 elements. A large subgroup ''H'' (preferably a maximal subgroup) of the Monster is selected in which it is easy to perform calculations. The subgroup ''H'' chosen is 3<sup>1+12</sup>.2.Suz.2, where Suz is the [[Suzuki group (mathematics)|Suzuki group]]. Elements of the monster are stored as words in the elements of ''H'' and an extra generator ''T''. It is reasonably quick to calculate the action of one of these words on a vector in ''V''. Using this action, it is possible to perform calculations (such as the order of an element of the monster). Wilson has exhibited vectors ''u'' and ''v'' whose joint stabilizer is the trivial group. Thus (for example) one can calculate the order of an element ''g'' of the monster by finding the smallest ''i'' &gt; 0 such that ''g''<sup>''i''</sup>''u'' = ''u'' and ''g''<sup>''i''</sup>''v'' = ''v''. This and similar constructions (in different [[characteristic (algebra)|characteristics]]) were used to find some of the non-local maximal subgroups of the monster group.