LMFDB - Abstract group 7920.a: Mathieu group $M

Show commands: Gap / Magma / SageMath

comment:Define the group as a permutation group

magma:G := PermutationGroup< 11 | (1,3,9,5,4)(2,6,7,10,8), (2,6,10,7)(3,9,4,5), (1,2,3,4,5,6,7,8,9,10,11) >;

gap:G := Group( (1,3,9,5,4)(2,6,7,10,8), (2,6,10,7)(3,9,4,5), (1,2,3,4,5,6,7,8,9,10,11) );

sage:G = PermutationGroup(['(1,3,9,5,4)(2,6,7,10,8)', '(2,6,10,7)(3,9,4,5)', '(1,2,3,4,5,6,7,8,9,10,11)'])

Group information

Description:	$M_{11}$
Order:	$7920$$\medspace = 2^{4} \cdot 3^{2} \cdot 5 \cdot 11 $	comment:Order of the group magma:Order(G); gap:Order(G); sage:G.order() sage_gap:G.Order()
Exponent:	$1320$$\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 11 $	comment:Exponent of the group magma:Exponent(G); gap:Exponent(G); sage:G.exponent() sage_gap:G.Exponent()
Automorphism group:	$M_{11}$, of order $7920$$\medspace = 2^{4} \cdot 3^{2} \cdot 5 \cdot 11 $	comment:Automorphism group gap:AutomorphismGroup(G); magma:AutomorphismGroup(G); sage_gap:G.AutomorphismGroup()
Composition factors:	$M_{11}$	comment:Composition factors of the group magma:CompositionFactors(G); gap:CompositionSeries(G); sage:G.composition_series() sage_gap:G.CompositionSeries()
Derived length:	$0$	comment:Derived length of the group magma:DerivedLength(G); gap:DerivedLength(G); sage_gap:G.DerivedLength()

This group is nonabelian and simple (hence nonsolvable, perfect, quasisimple, and almost simple).

comment:Determine if the group G is abelian

magma:IsAbelian(G);

gap:IsAbelian(G);

sage:G.is_abelian()

sage_gap:G.IsAbelian()

comment:Determine if the group G is cyclic

magma:IsCyclic(G);

gap:IsCyclic(G);

sage:G.is_cyclic()

sage_gap:G.IsCyclic()

comment:Determine if the group G is nilpotent

magma:IsNilpotent(G);

gap:IsNilpotentGroup(G);

sage:G.is_nilpotent()

sage_gap:G.IsNilpotentGroup()

comment:Determine if the group G is solvable

magma:IsSolvable(G);

gap:IsSolvableGroup(G);

sage:G.is_solvable()

sage_gap:G.IsSolvableGroup()

comment:Determine if the group G is supersolvable

gap:IsSupersolvableGroup(G);

sage:G.is_supersolvable()

sage_gap:G.IsSupersolvableGroup()

comment:Determine if the group G is simple

magma:IsSimple(G);

gap:IsSimpleGroup(G);

sage_gap:G.IsSimpleGroup()

Group statistics

comment:Compute statistics for the group G

magma:// Magma code to output the first two rows of the group statistics table element_orders := [Order(g) : g in G]; orders := Set(element_orders); printf "Orders: %o\n", orders; printf "Elements: %o %o\n", [#[x : x in element_orders | x eq n] : n in orders], Order(G); cc_orders := [cc[1] : cc in ConjugacyClasses(G)]; printf "Conjugacy classes: %o %o\n", [#[x : x in cc_orders | x eq n] : n in orders], #cc_orders;

gap:# Gap code to output the first two rows of the group statistics table element_orders := List(Elements(G), g -> Order(g)); orders := Set(element_orders); Print("Orders: ", orders, "\n"); element_counts := List(orders, n -> Length(Filtered(element_orders, x -> x = n))); Print("Elements: ", element_counts, " ", Size(G), "\n"); cc_orders := List(ConjugacyClasses(G), cc -> Order(Representative(cc))); cc_counts := List(orders, n -> Length(Filtered(cc_orders, x -> x = n))); Print("Conjugacy classes: ", cc_counts, " ", Length(ConjugacyClasses(G)), "\n");

sage:# Sage code to output the first two rows of the group statistics table element_orders = [g.order() for g in G] orders = sorted(list(set(element_orders))) print("Orders:", orders) print("Elements:", [element_orders.count(n) for n in orders], G.order()) cc_orders = [cc[0].order() for cc in G.conjugacy_classes()] print("Conjugacy classes:", [cc_orders.count(n) for n in orders], len(cc_orders))

Order	1	2	3	4	5	6	8	11
Elements	1	165	440	990	1584	1320	1980	1440	7920
Conjugacy classes	1	1	1	1	1	1	2	2	10
Divisions	1	1	1	1	1	1	1	1	8
Autjugacy classes	1	1	1	1	1	1	2	2	10

comment:Compute statistics about the characters of G

magma:// Outputs [<d_1,c_1>, <d_2,c_2>, ...] where c_i is the number of irr. complex chars. of G with degree d_i CharacterDegrees(G);

gap:# Outputs [[d_1,c_1], [d_2,c_2], ...] where c_i is the number of irr. complex chars. of G with degree d_i CharacterDegrees(G);

sage:# Outputs [[d_1,c_1], [d_2,c_2], ...] where c_i is the number of irr. complex chars. of G with degree d_i character_degrees = [c[0] for c in G.character_table()] [[n, character_degrees.count(n)] for n in set(character_degrees)]

sage_gap:G.CharacterDegrees()

Dimension	1	10	11	16	20	32	44	45	55
Irr. complex chars.	1	3	1	2	0	0	1	1	1	10
Irr. rational chars.	1	1	1	0	1	1	1	1	1	8

Minimal presentations

Permutation degree:	$11$
Transitive degree:	$11$
Rank:	$2$
Inequivalent generating pairs:	$6478$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	10	10	10
Arbitrary	10	10	10

Constructions

Show commands: Gap / Magma / SageMath

Permutation group:	Degree $11$ $\langle(1,3,9,5,4)(2,6,7,10,8), (2,6,10,7)(3,9,4,5), (1,2,3,4,5,6,7,8,9,10,11)\rangle$ (1,3,9,5,4)(2,6,7,10,8), (2,6,10,7)(3,9,4,5), (1,2,3,4,5,6,7,8,9,10,11)
comment:Define the group as a permutation group magma:G := PermutationGroup< 11 \| (1,3,9,5,4)(2,6,7,10,8), (2,6,10,7)(3,9,4,5), (1,2,3,4,5,6,7,8,9,10,11) >; gap:G := Group( (1,3,9,5,4)(2,6,7,10,8), (2,6,10,7)(3,9,4,5), (1,2,3,4,5,6,7,8,9,10,11) ); sage:G = PermutationGroup(['(1,3,9,5,4)(2,6,7,10,8)', '(2,6,10,7)(3,9,4,5)', '(1,2,3,4,5,6,7,8,9,10,11)'])
Transitive group:	11T6	12T272	22T22	more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	not isomorphic to a non-trivial semidirect product
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Aut. group:	$\Aut(C_2\times M_{11})$

Elements of the group are displayed as permutations of degree 11.

Homology

Abelianization:	$C_1 $	comment:The abelianization of the group magma:quo< G \| CommutatorSubgroup(G) >; gap:FactorGroup(G, DerivedSubgroup(G)); sage:G.quotient(G.commutator())
Schur multiplier:	$C_1$	comment:The Schur multiplier of the group gap:AbelianInvariantsMultiplier(G); sage:G.homology(2) sage_gap:G.AbelianInvariantsMultiplier()
Commutator length:	$1$	comment:The commutator length of the group gap:CommutatorLength(G); sage_gap:G.CommutatorLength()

Subgroups

comment:List of subgroups of the group

magma:Subgroups(G);

gap:AllSubgroups(G);

sage:G.subgroups()

sage_gap:G.AllSubgroups()

There are 8651 subgroups in 39 conjugacy classes, 2 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $M_{11}$	comment:Center of the group magma:Center(G); gap:Center(G); sage:G.center() sage_gap:G.Center()
Commutator:	$G' \simeq$ $M_{11}$	$G/G' \simeq$ $C_1$	comment:Commutator subgroup of the group G magma:CommutatorSubgroup(G); gap:DerivedSubgroup(G); sage:G.commutator() sage_gap:G.DerivedSubgroup()
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $M_{11}$	comment:Frattini subgroup of the group G magma:FrattiniSubgroup(G); gap:FrattiniSubgroup(G); sage:G.frattini_subgroup() sage_gap:G.FrattiniSubgroup()
Fitting:	$\operatorname{Fit} \simeq$ $C_1$	$G/\operatorname{Fit} \simeq$ $M_{11}$	comment:Fitting subgroup of the group G magma:FittingSubgroup(G); gap:FittingSubgroup(G); sage:G.fitting_subgroup() sage_gap:G.FittingSubgroup()
Radical:	$R \simeq$ $C_1$	$G/R \simeq$ $M_{11}$	comment:Radical of the group G magma:Radical(G); gap:SolvableRadical(G); sage_gap:G.SolvableRadical()
Socle:	$\operatorname{soc} \simeq$ $M_{11}$	$G/\operatorname{soc} \simeq$ $C_1$	comment:Socle of the group G magma:Socle(G); gap:Socle(G); sage:G.socle() sage_gap:G.Socle()
2-Sylow subgroup:	$P_{ 2 } \simeq$ $\SD_{16}$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$
11-Sylow subgroup:	$P_{ 11 } \simeq$ $C_{11}$

Subgroup diagram and profile

For the default diagram, subgroups are sorted vertically by the number of prime divisors (counted with multiplicity) in their orders.
To see subgroups sorted vertically by order instead, check this box.

Subgroup information

Click on a subgroup in the diagram to see information about it.

See a full page version of the diagram

Series

Derived series

$M_{11}$

comment:Derived series of the group GF

magma:DerivedSeries(G);

gap:DerivedSeriesOfGroup(G);

sage:G.derived_series()

sage_gap:G.DerivedSeriesOfGroup()

Chief series

$M_{11}$

$\rhd$

$C_1$

comment:Chief series of the group G

magma:ChiefSeries(G);

gap:ChiefSeries(G);

sage_gap:G.ChiefSeries()

Lower central series

$M_{11}$

comment:The lower central series of the group G

magma:LowerCentralSeries(G);

gap:LowerCentralSeriesOfGroup(G);

sage:G.lower_central_series()

sage_gap:G.LowerCentralSeriesOfGroup()

Upper central series

$C_1$

comment:The upper central series of the group G

magma:UpperCentralSeries(G);

gap:UpperCentralSeriesOfGroup(G);

sage:G.upper_central_series()

sage_gap:G.UpperCentralSeriesOfGroup()

Supergroups

This group is a maximal subgroup of 10 larger groups in the database.

This group is a maximal quotient of 6 larger groups in the database.

Character theory

comment:Character table

magma:CharacterTable(G); // Output not guaranteed to exactly match the LMFDB table

gap:CharacterTable(G); # Output not guaranteed to exactly match the LMFDB table

sage:G.character_table() # Output not guaranteed to exactly match the LMFDB table

sage_gap:G.CharacterTable() # Output not guaranteed to exactly match the LMFDB table

Complex character table

		1A	2A	3A	4A	5A	6A	8A1	8A-1	11A1	11A-1
	Size	1	165	440	990	1584	1320	990	990	720	720
	2 P	1A	1A	3A	2A	5A	3A	4A	4A	11A-1	11A1
	3 P	1A	2A	1A	4A	5A	2A	8A1	8A-1	11A1	11A-1
	5 P	1A	2A	3A	4A	1A	6A	8A-1	8A1	11A1	11A-1
	11 P	1A	2A	3A	4A	5A	6A	8A1	8A-1	1A	1A
	Type
7920.a.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
7920.a.10a	R	$10$	$2$	$1$	$2$	$0$	$- 1$	$0$	$0$	$- 1$	$- 1$
7920.a.10b1	C	$10$	$- 2$	$1$	$0$	$0$	$1$	$- ζ_{8} - ζ_{8}^{3}$	$ζ_{8} + ζ_{8}^{3}$	$- 1$	$- 1$
7920.a.10b2	C	$10$	$- 2$	$1$	$0$	$0$	$1$	$ζ_{8} + ζ_{8}^{3}$	$- ζ_{8} - ζ_{8}^{3}$	$- 1$	$- 1$
7920.a.11a	R	$11$	$3$	$2$	$- 1$	$1$	$0$	$- 1$	$- 1$	$0$	$0$
7920.a.16a1	C	$16$	$0$	$- 2$	$0$	$1$	$0$	$0$	$0$	$- ζ_{11}^{- 2} - 1 - ζ_{11} - ζ_{11}^{3} - ζ_{11}^{4} - ζ_{11}^{5}$	$ζ_{11}^{- 2} + ζ_{11} + ζ_{11}^{3} + ζ_{11}^{4} + ζ_{11}^{5}$
7920.a.16a2	C	$16$	$0$	$- 2$	$0$	$1$	$0$	$0$	$0$	$ζ_{11}^{- 2} + ζ_{11} + ζ_{11}^{3} + ζ_{11}^{4} + ζ_{11}^{5}$	$- ζ_{11}^{- 2} - 1 - ζ_{11} - ζ_{11}^{3} - ζ_{11}^{4} - ζ_{11}^{5}$
7920.a.44a	R	$44$	$4$	$- 1$	$0$	$- 1$	$1$	$0$	$0$	$0$	$0$
7920.a.45a	R	$45$	$- 3$	$0$	$1$	$0$	$0$	$- 1$	$- 1$	$1$	$1$
7920.a.55a	R	$55$	$- 1$	$1$	$- 1$	$0$	$- 1$	$1$	$1$	$0$	$0$

Rational character table

		1A	2A	3A	4A	5A	6A	8A	11A
	Size	1	165	440	990	1584	1320	1980	1440
	2 P	1A	1A	3A	2A	5A	3A	4A	11A
	3 P	1A	2A	1A	4A	5A	2A	8A	11A
	5 P	1A	2A	3A	4A	1A	6A	8A	11A
	11 P	1A	2A	3A	4A	5A	6A	8A	1A
7920.a.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
7920.a.10a		$10$	$2$	$1$	$2$	$0$	$- 1$	$0$	$- 1$
7920.a.10b		$20$	$- 4$	$2$	$0$	$0$	$2$	$0$	$- 2$
7920.a.11a		$11$	$3$	$2$	$- 1$	$1$	$0$	$- 1$	$0$
7920.a.16a		$32$	$0$	$- 4$	$0$	$2$	$0$	$0$	$- 1$
7920.a.44a		$44$	$4$	$- 1$	$0$	$- 1$	$1$	$0$	$0$
7920.a.45a		$45$	$- 3$	$0$	$1$	$0$	$0$	$- 1$	$1$
7920.a.55a		$55$	$- 1$	$1$	$- 1$	$0$	$- 1$	$1$	$0$

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Label	7920.a
Order	\( 2^{4} \cdot 3^{2} \cdot 5 \cdot 11 \)
Exponent	\( 2^{3} \cdot 3 \cdot 5 \cdot 11 \)

Simple	yes
$\card{G^{\mathrm{ab}}}$	\( 1 \)
$\card{Z(G)}$	\( 1 \)
$\card{\Aut(G)}$	\( 2^{4} \cdot 3^{2} \cdot 5 \cdot 11 \)
$\card{\mathrm{Out}(G)}$	\( 1 \)
Perm deg.	$11$
Trans deg.	$11$
Rank	$2$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

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Group information

Group statistics

Minimal presentations

Minimal degrees of faithful linear representations

Constructions

Homology

Subgroups

Special subgroups

Subgroup diagram and profile

Classes of subgroups up to conjugation

Classes of subgroups up to automorphism

Normal subgroups

Normal subgroups up to automorphism

Subgroup information

Classes of subgroups up to conjugation

Classes of subgroups up to automorphism

Normal subgroups (quotient in parentheses)

Normal subgroups up to automorphism (quotient in parentheses)

Series

Supergroups

Character theory

Complex character table

Rational character table

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.