represents the sporadic simple Mathieu group .
MathieuGroupM22
represents the sporadic simple Mathieu group .
Details
- By default, MathieuGroupM22 [] is represented as a permutation group acting on points {1,…,22}.
Background & Context
- MathieuGroupM22 [] represents the Mathieu group , which is a group of order . It is one of the 26 sporadic simple groups of finite order. The default representation of MathieuGroupM22 is as a permutation group on the points having two generators.
- The Mathieu group is the fourth smallest of the sporadic finite simple groups. It was discovered (along with the other four Mathieu groups MathieuGroupM11 , MathieuGroupM12 , MathieuGroupM23 and MathieuGroupM24 ) by mathematician Émile Léonard Mathieu in the late 1800s, making these groups tied for first in chronological order of discovery among sporadic groups. MathieuGroupM22 is 3-transitive in the sense that there exists at least one group element mapping any unique 3-tuple of elements of MathieuGroupM22 to any other unique 3-tuple therein. In addition to its permutation representation, can be defined in terms of generators and relations as and has a 3-transitive representation on , the point stabilizer of which is the projective special linear group . Along with the other sporadic simple groups, the Mathieu groups played a foundational role in the monumental (and complete) classification of finite simple groups.
- The usual group theoretic functions may be applied to MathieuGroupM22 [], including GroupOrder , GroupGenerators , GroupElements and so on. A number of precomputed properties of the Mathieu group are available via FiniteGroupData [{"Mathieu",22},"prop"].
- MathieuGroupM22 is related to a number of other symbols. Along with MathieuGroupM11 , MathieuGroupM12 , MathieuGroupM23 and MathieuGroupM24 , MathieuGroupM22 is one of five groups cumulatively referred to as the so-called "first generation" of sporadic finite simple groups. It is also one of 20 so-called "happy" sporadic groups, which all appear as a subquotient of the monster group.
Examples
Basic Examples (1)
Order of the group :
Generators of a permutation representation of the group :
Tech Notes
Related Guides
History
Text
Wolfram Research (2010), MathieuGroupM22, Wolfram Language function, https://reference.wolfram.com/language/ref/MathieuGroupM22.html.
CMS
Wolfram Language. 2010. "MathieuGroupM22." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/MathieuGroupM22.html.
APA
Wolfram Language. (2010). MathieuGroupM22. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/MathieuGroupM22.html
BibTeX
@misc{reference.wolfram_2025_mathieugroupm22, author="Wolfram Research", title="{MathieuGroupM22}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/MathieuGroupM22.html}", note=[Accessed: 05-December-2025]}
BibLaTeX
@online{reference.wolfram_2025_mathieugroupm22, organization={Wolfram Research}, title={MathieuGroupM22}, year={2010}, url={https://reference.wolfram.com/language/ref/MathieuGroupM22.html}, note=[Accessed: 05-December-2025]}