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Five sporadic simple groups
In group theory, a topic in abstract algebra, the Mathieu groups are the five sporadic simple groups M11, M12, M22, M23 and M24 introduced by Émile Mathieu (1861
Mathieu_group
Sporadic simple group
In the area of modern algebra known as group theory, the Mathieu group M24 is a sporadic simple group of order 244,823,040 = 210 · 33 · 5 · 7 · 11 ·
Mathieu_group_M24
Sporadic simple group
In the area of modern algebra known as group theory, the Mathieu group M11 is a sporadic simple group of order 7,920 = 11 · 10 · 9 · 8 = 24 · 32 · 5 ·
Mathieu_group_M11
Sporadic simple group
In the area of modern algebra known as group theory, the Mathieu group M23 is a sporadic simple group of order 10,200,960 = 27 · 32 · 5 · 7 · 11 · 23
Mathieu_group_M23
Sporadic simple group
In the area of modern algebra known as group theory, the Mathieu group M12 is a sporadic simple group of order 95,040 = 12 · 11 · 10 · 9 · 8 = 26 ·
Mathieu_group_M12
Sporadic simple group
In the area of modern algebra known as group theory, the Mathieu group M22 is a sporadic simple group of order 443,520 = 27 · 32 · 5 · 7 · 11 ≈ 4×105
Mathieu_group_M22
Construction in group theory
the Mathieu group M24, one of the sporadic simple groups; in this context, one refers to PSL(3, 4) as M21, though it is not properly a Mathieu group itself
Projective_linear_group
Topic in group theory and harmonic analysis (Niemeier lattice-mock theta connection)
functions. It is a generalization of the Mathieu moonshine phenomenon connecting representations of the Mathieu group M24 with K3 surfaces. The usage of the
Umbral_moonshine
Block design in combinatorial mathematics
to group theory. In particular, the finite simple groups called Mathieu groups arise as automorphism groups of Steiner systems: The Mathieu group M11
Steiner_system
the automorphism group of the projective geometry of a vector space. The group PΓL(3,4) can be used to construct the Mathieu group M24, which is one
Semilinear_map
Groupoid related to the Mathieu group M12
mathematics, the Mathieu groupoid M13 is a groupoid acting on 13 points such that the stabilizer of each point is the Mathieu group M12. It was introduced
Mathieu_groupoid
Automorphism group of the Klein quartic
subgroup of the Mathieu group M21, the subgroup acts non-transitively on 21 points. (Richter) Richter, David A., How to Make the Mathieu Group M24, archived
PSL(2,7)
Natural number
and angle trisector. The Mathieu group M 11 {\displaystyle \mathrm {M} _{11}} is the smallest of twenty-six sporadic groups. It has order 7920 = 2 4 ⋅
11_(number)
Croatian mathematician (1932–2022)
of the Janko groups, sporadic simple groups in group theory. The first few sporadic simple groups were discovered by Émile Léonard Mathieu, which were
Zvonimir_Janko
Finite simple group type not classified as Lie, cyclic or alternating
sporadic groups were discovered by Émile Mathieu in the 1860s and the other twenty-one were found between 1965 (J1) and 1975 (J4). Several of these groups were
Sporadic_group
Monster and modular connection
T14C into representations of 2.A7 T11A into representations of the Mathieu group 2.M12 Queen found that the traces of non-identity elements also yielded
Monstrous_moonshine
Dutch cyclist
Mathieu van der Poel (born 19 January 1995), nicknamed "The Flying Dutchman", is a Belgian-born Dutch professional cyclist who currently rides for the
Mathieu_van_der_Poel
Name list
Mathieu is both a surname and a given name. Notable people with the name include: André Mathieu (1929–1968), Canadian pianist and composer Anselme Mathieu
Mathieu
Type of linear error-correcting code
invariant), is the Mathieu group M 23 {\displaystyle M_{23}} . The automorphism group of the extended binary Golay code is the Mathieu group M 24 {\displaystyle
Binary_Golay_code
Natural number
22 appears prominently within sporadic groups. The Mathieu group M22 is one of 26 sporadic finite simple groups, defined as the 3-transitive permutation
22_(number)
Natural number
design. The automorphism group of the extended binary Golay code, equivalently of this Steiner system, is the Mathieu group M 24 {\displaystyle M_{24}}
24_(number)
groups states that every finite simple group is cyclic, or alternating, or in one of 16 families of groups of Lie type, or one of 26 sporadic groups.
List_of_finite_simple_groups
Theorem classifying finite simple groups
of finite simple groups (popularly called the enormous theorem) is a result of group theory stating that every finite simple group is either cyclic,
Classification of finite simple groups
Classification_of_finite_simple_groups
Natural number
The five Mathieu groups constitute the first generation in the happy family of sporadic groups. These are also the first five sporadic groups to have been
5
Natural number
{\displaystyle \mathbb {B} _{23}} has Mathieu group M 23 {\displaystyle \mathbb {M} _{23}} as its automorphism group, which is the second largest member
23_(number)
PSU3(2n) The Suzuki groups Sz(2n) The Tits group 2F4(2)' The Steinberg group 3D4(2) The Mathieu group M11 The Janko group J1 Quasithin group Aschbacher, Michael
Thin group (finite group theory)
Thin_group_(finite_group_theory)
Mathematical tool
is a mathematical tool introduced by Rob T. Curtis for studying the Mathieu groups, binary Golay code and Leech lattice. The Miracle Octad Generator is
Miracle_Octad_Generator
Natural number
the order of the friendly giant, the largest sporadic group: five first generation Mathieu groups, seven second generation subquotients of the Leech lattice
6
French singer (born 1946)
Mireille Mathieu (French pronunciation: [miʁɛj matjø] ; born 22 July 1946) is a French singer. She has recorded over 1,200 songs in eleven languages,
Mireille_Mathieu
Natural number
951212 = 9048004641; 90480 + 04641 = 95121 95,040 = the order of the Mathieu group M12 95,420 = number of 22-bead binary necklaces with beads of 2 colors
90,000
24-dimensional repeating pattern of points
group Co1, a finite simple group, and one of the sporadic groups. Many other sporadic groups, such as the remaining Conway groups and Mathieu groups,
Leech_lattice
French investment banker
company to Czech businessman Daniel Křetínský. Le Monde's Independency Group, a minority shareholder that aims to protect the paper's editorial independence
Matthieu_Pigasse
Transformations induced by a mathematical group
This is useful, for instance, in studying the action of the large Mathieu group on a 24-set and in studying symmetry in certain models of finite geometries
Group_action
French mathematician
mathematical physics. He has given his name to the Mathieu functions, Mathieu groups and Mathieu transformation. He authored a treatise of mathematical
Émile_Léonard_Mathieu
Index of articles associated with the same name
known as group theory, the Janko groups are the four sporadic simple groups J1, J2, J3 and J4 introduced by Zvonimir Janko. Unlike the Mathieu groups, Conway
Janko_group
Group whose operation is composition of permutations
Permutation Groups in 1964. 2-transitive group Rank 3 permutation group Mathieu group The notations SM and SM are also used. Rotman 2006, p. 148, Definition
Permutation_group
1898 Miller published a paper incorrectly claiming to prove that the Mathieu group M24 does not exist, though in 1900 he pointed out that his proof was
List_of_incomplete_proofs
Four finite groups derived from the Leech lattice
code (as diagonal matrices with 1 or −1 as diagonal elements) by the Mathieu group M24 (as permutation matrices). N ≈ 212:M24. A standard representation
Conway_group
Topics referred to by the same term
star cluster also known as the Wild Duck Cluster Mathieu group M11 in the mathematical field of group theory Ritter M11 UltraClave, a model of autoclave
M11
Unsolved problem in mathematics
solvable group is realizable over Q {\displaystyle \mathbb {Q} } . It is also known that every simple sporadic group, except possibly the Mathieu group M23
Inverse_Galois_problem
Strongly regular graph
automorphism group is the Mathieu group M22. Cameron graph Higman–Sims graph Gewirtz graph "Mesner graph with parameters (77,16,0,4). The automorphism group is
M22_graph
German mathematician
2-subgroup of the Mathieu group M24 on 24 letters. Shortly afterwards Graham Higman and John McKay demonstrated that such a group exists, using a computer
Dieter_Held
endpoints of the edges that bisect the squares and octahedra) to yield the Mathieu group M24. Compound of five small cubicuboctahedra List of uniform polyhedra
Small_cubicuboctahedron
Finite group
quasidihedral: PSL3(Fq) for q ≡ 3 mod 4, PSU3(Fq) for q ≡ 1 mod 4, the Mathieu group M11, GL2(Fq) for q ≡ 3 mod 4. Dummit, D. S.; Foote, R. (2004). Abstract
Quasidihedral_group
Set with associative invertible operation
related to the appearance of Goldstone bosons. Finite symmetry groups such as the Mathieu groups are used in coding theory, which is in turn applied in error
Group_(mathematics)
Group without normal subgroups other than the trivial group and itself
groups). These groups (the groups of Lie type, together with the cyclic groups, alternating groups, and the five exceptional Mathieu groups) were believed
Simple_group
Sporadic simple group
maximal subgroups isomorphic to the Mathieu group M22. An outer automorphism interchanges the two classes of M22 groups. This outer automorphism is realized
McLaughlin_sporadic_group
French footballer (born 1984)
Mathieu Valbuena (born 28 September 1984) is a French professional footballer who plays as an attacking midfielder for Maltese Challenge League club St
Mathieu_Valbuena
Concept in group theory
there is a group element g in G which maps ai to bi for each i between 1 and k. The Mathieu groups are important examples. Every group is trivially
Multiply transitive group action
Multiply_transitive_group_action
Mathematical connection between field theory and group theory
(Mathieu group M23) of the 26 sporadic simple groups. There is even a polynomial with integral coefficients whose Galois group is the Monster group. In
Galois_theory
Theory of subatomic structure
Tachikawa discovered connections between a different sporadic group, the Mathieu group M24, and a certain version[which?] of string theory. Miranda Cheng
String_theory
French politician (born 1986)
French-Israeli Parliamentary Friendship Group from 2023 to 2024. "Législatives dans le Val-de-Marne : Mathieu Lefèvre, seul nouveau visage de la majorité
Mathieu_Lefèvre
Compact Riemann surface of genus 3
The automorphism group can be augmented (by a symmetry which is not realized by a symmetry of the tiling) to yield the Mathieu group M24. Corresponding
Klein_quartic
Topics referred to by the same term
Manhattan BFW M.23, two-seater sports plane The Mathieu group M23 in the mathematical field of group theory m23 software distribution system, a Linux
M23
Aspect of mathematical group theory
exceptional itself. The full automorphism group of A6 appears naturally as a maximal subgroup of the Mathieu group M12 in 2 ways, as either a subgroup fixing
Automorphisms of the symmetric and alternating groups
Automorphisms_of_the_symmetric_and_alternating_groups
French footballer (born 1984)
Mathieu Pierre Flamini (born 7 March 1984) is a French entrepreneur and former professional footballer. A midfielder, he played for French side Marseille
Mathieu_Flamini
Concept in incidence geometry
1990s. Some sporadic simple groups, for example the Hall-Janko group and the Mathieu groups, act as automorphism groups of near polygons. A near 2d-gon
Near_polygon
Slovenian cyclist (born 1998)
peloton with a select group of riders. He accelerated twice more on the final ascent of Oude Kwaremont and the Paterberg and only Mathieu van der Poel was
Tadej_Pogačar
related to the Leech lattice. The Mathieu group M 24 {\displaystyle M_{24}} , one of the sporadic simple groups, is the group of automorphisms of the extended
Exceptional_object
Sporadic simple group
check for other rank 3 permutation groups on 100 points. They soon focused on a possible one containing the Mathieu group M22, which has permutation representations
Higman–Sims_group
FIFA World Cup group
Group B of the 2026 FIFA World Cup took place from June 12 to 24, 2026. The group consisted of Canada (co-host), Bosnia and Herzegovina, Qatar, and Switzerland
2026_FIFA_World_Cup_Group_B
Strongly regular graph
It is one of a small number of strongly regular graphs on which the Mathieu group M22 acts as symmetries taking every vertex to every other vertex. The
Cameron_graph
Natural number
number of objects that are permuted by the series of five finite, simple Mathieu groups M n {\displaystyle \mathbb {M} _{n}} (collectively), as defined by permutations
92_(number)
French association football player (born 1985)
Mathieu Debuchy (French pronunciation: [matjø dəbyʃi]; born 28 July 1985) is a French former professional footballer who played as a right-back. Debuchy
Mathieu_Debuchy
Topics referred to by the same term
movement in the People's Republic of Congo. The Mathieu group M24 in the mathematical field of group theory M-24 (Michigan highway), a state highway in
M24
French fashion designer (1936–2008)
Yves Henri Donat Mathieu-Saint-Laurent (1 August 1936 – 1 June 2008), better known as Yves Saint Laurent (/ˌiːv ˌsæ̃ lɔːˈrɒ̃/, also UK: /- lɒ-/, US: /-
Yves_Saint_Laurent_(designer)
Category where every morphism is invertible; generalization of a group
point form a copy of the Mathieu group M12. If a groupoid has only one object, then the set of its morphisms forms a group. Using the algebraic definition
Groupoid
Topics referred to by the same term
of IEC metric screw sized electrical connector Mathieu group M12, in the mathematical field of group theory Magic 2012, the thirteenth core set in Magic:
M12
Pair of related error-correcting codes
The automorphism group of the (original) ternary Golay code is the Mathieu group M11, which is the smallest of the sporadic simple groups. The complete weight
Ternary_Golay_code
projective special unitary groups over a finite field of odd order, depending on a certain congruence, or to the Mathieu group M 11 {\displaystyle M_{11}}
Alperin–Brauer–Gorenstein theorem
Alperin–Brauer–Gorenstein_theorem
3-transpositions in the symmetry group Fi22 of the graph. The Fischer groups are named by analogy with the large Mathieu groups. In Fi22 a maximal set of 3-transpositions
Fischer_group
Natural number
7919 – 1000th prime number 7920 – the order of the Mathieu group M11, the smallest sporadic simple group 7921 = 892, centered octagonal number 7944 – nonagonal
7000_(number)
linear group Group of Lie type Group scheme HN group Janko group Lie group Simple Lie group Linear algebraic group List of finite simple groups Mathieu group
List_of_group_theory_topics
unitary group U3(3), the alternating group A7, the Mathieu group M11, and the Tits group. (The Tits group was overlooked in Thomson's original announcement
N-group_(finite_group_theory)
English mathematician (1937–2020)
connections to string theory. Conway introduced the Mathieu groupoid, an extension of the Mathieu group M12 to 13 points. As a graduate student, he proved
John_Horton_Conway
Mathematical group
that other finite simple groups exist (for example, Mathieu groups), gradually a belief formed that nearly all finite simple groups can be accounted for by
Group_of_Lie_type
2017 American film
footage supernatural horror film directed by Victor Mathieu from a screenplay written by Mathieu, Corbin Billings and Shariya Lynn. The film stars Justin
The_Monster_Project
Sporadic simple group
or 24 then it is contained in either Z/2Z × Co2 or Z/2Z × Co3. The Mathieu group M23 is isomorphic to a maximal subgroup of Co2 and one representation
Conway_group_Co2
Type of smooth complex surface of kodaira dimension 0
Enriques surface Tate conjecture Mathieu moonshine, a mysterious relationship between K3 surfaces and the Mathieu group M24. Huybrechts (2016), Remark 1
K3_surface
French worker and illegalist anarchist
gang, a closely linked and similar group. Following the arrest of some of the Intransigents and Schouppe, Mathieu moved to the northern districts of Paris
Gustave_Mathieu
Sporadic simple group
special group 21+6 by the linear group L3(2), which is the same involution centralizer as the Mathieu group M24. A second such group is the linear group L5(2)
Held_group
1988 mathematical book
The interrelated topics featured in the book include Golay codes, Mathieu groups and Monstrous Moonshine. Many of the chapters are updates of papers
Sphere Packings, Lattices and Groups
Sphere_Packings,_Lattices_and_Groups
Topics referred to by the same term
cryoprotectant, used in cryonics and cryopreservation The Mathieu group M22 in the mathematical field of group theory M22 graph, in graph theory Messier 22, a globular
M22
Career achievements of cyclist Mathieu van der Poel
Mathieu van der Poel is a Dutch professional racing cyclist for UCI WorldTeam Alpecin–Premier Tech, competing in the cyclo-cross, road bicycle racing
List of career achievements by Mathieu van der Poel
List_of_career_achievements_by_Mathieu_van_der_Poel
French novelist (born 1978)
Nicolas Mathieu (French pronunciation: [nikɔla matjø]; born 2 June 1978 in Épinal) is a French novelist who has also written short fiction and children's
Nicolas_Mathieu_(writer)
automorphism group of A6 is M10, the Mathieu group of degree 10, whose Schur cover is a triple cover. The Schur covers of the symmetric group S6 itself have
Covering groups of the alternating and symmetric groups
Covering_groups_of_the_alternating_and_symmetric_groups
French footballer (born 1983)
Jérémy Mathieu (born 29 October 1983) is a French former professional footballer who played as a centre-back or a left-back. Mathieu made 172 Ligue 1
Jérémy_Mathieu
Cycling race
warning of an ongoing heatwave in the Corrèze department. It was won by Mathieu van der Poel (Alpecin–Premier Tech) from a breakaway. After a rest day
2026_Tour_de_France
with trivial right nucleus. Also, using an exact factorization of the Mathieu group M24, it is possible to construct a non-Moufang simple Bol loop which
List of problems in loop theory and quasigroup theory
List_of_problems_in_loop_theory_and_quasigroup_theory
Theorem in algebraic geometry
2012-03-05, retrieved 2015-09-04 Richter, David A., How to Make the Mathieu Group M24, archived from the original on 2010-01-16, retrieved 2010-04-15
Hurwitz's automorphisms theorem
Hurwitz's_automorphisms_theorem
Higman–Sims group Janko groups Lyons group The Mathieu groups McLaughlin group Monster group O'Nan group Rudvalis group Suzuki sporadic group Thompson group Counterexample
List_of_mathematical_examples
\mathrm {A} _{6}.} Two other groups, the Mathieu group M 10 {\displaystyle \mathrm {M} _{10}} and the projective general linear group PGL 2 ( 9 ) {\displaystyle
Almost_simple_group
Football tournament qualification stage
9 July 2026 (2026-07-09) 20:00 Karađorđe Stadium, Novi Sad Attendance: 6,557 Referee: Mathieu Vernice (France) 16 July 2026 (2026-07-16) 20:15 Ferencváros Stadion, Budapest
2026–27 UEFA Europa League qualifying
2026–27_UEFA_Europa_League_qualifying
Polynomial that permutes a ring
GF(qr) form a group under the operation of composition modulo x q r − x {\displaystyle x^{q^{r}}-x} , which is known as the Betti-Mathieu group, isomorphic
Permutation_polynomial
French musical group
Kid Francescoli is an electropop project by French musician Mathieu Hocine. The project was founded by Hocine in 2002 in Marseille. The name is a tribute
Kid_Francescoli
French singer-songwriter
Mathieu Blanc-Francard, known professionally as Sinclair (born 19 July 1970), is a French musician and singer-songwriter. Born in 1970 at Tours, France
Sinclair_(singer)
Algebraic structure
Sp6(2) The alternating groups on 5, 6, 8, 9 points PSL2(p) for p a Fermat or Mersenne prime, Lε 3(3), Lε 4(3), G2(3) The Mathieu groups M11, M12, M22, M23
Quasithin_group
French water polo player (born 1982)
was part of the French team at the 2016 Summer Olympics, where the team was eliminated in the group stage. "Mathieu PEISSON - Olympic - France". v t e
Mathieu_Peisson
French rock group
and Mathieu "Phil" Jourdain on the drums. Both are Mont-de-Marsan natives and former members of the band Wolfunkind [fr]. The name of the group was proposed
The_Inspector_Cluzo
French singer-songwriter
Mathieu Boogaerts (born 1970) is a French singer-songwriter. The son of a pharmacist mother and antiquarian father, Mathieu was born in Fontenay-sous-Bois
Mathieu_Boogaerts
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