LMFDB - Abstract group 5040.w: $S

Show commands: Gap / Magma / SageMath

magma:G := SymmetricGroup(7);

gap:G := SymmetricGroup(7);

sage:G = SymmetricGroup(7)

comment:Define the group as a permutation group

Group information

Description:	$S_7$
Order:	$5040$$\medspace = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 $	comment:Order of the group magma:Order(G); gap:Order(G); sage:G.order() sage_gap:G.Order()
Exponent:	$420$$\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 7 $	comment:Exponent of the group magma:Exponent(G); gap:Exponent(G); sage:G.exponent() sage_gap:G.Exponent()
Automorphism group:	$S_7$, of order $5040$$\medspace = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 $	comment:Automorphism group gap:AutomorphismGroup(G); magma:AutomorphismGroup(G); sage_gap:G.AutomorphismGroup()
Composition factors:	$C_2$, $A_7$	comment:Composition factors of the group magma:CompositionFactors(G); gap:CompositionSeries(G); sage:G.composition_series() sage_gap:G.CompositionSeries()
Derived length:	$1$	comment:Derived length of the group magma:DerivedLength(G); gap:DerivedLength(G); sage_gap:G.DerivedLength()

This group is nonabelian, almost simple, nonsolvable, and rational.

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	7	10	12
Elements	1	231	350	840	504	1470	720	504	420	5040
Conjugacy classes	1	3	2	2	1	3	1	1	1	15
Divisions	1	3	2	2	1	3	1	1	1	15
Autjugacy classes	1	3	2	2	1	3	1	1	1	15

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	6	14	15	20	21	35
Irr. complex chars.	2	2	4	2	1	2	2	15
Irr. rational chars.	2	2	4	2	1	2	2	15

Minimal presentations

Permutation degree:	$7$
Transitive degree:	$7$
Rank:	$2$
Inequivalent generating pairs:	$3090$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	6	6	6
Arbitrary	6	6	6

Constructions

Show commands: Gap / Magma / SageMath

Permutation group:	$\langle(1,2,3,4,5,6,7), (1,2)\rangle$ (1,2,3,4,5,6,7), (1,2)
comment:Define the group as a permutation group magma:G := PermutationGroup< 7 \| (1,2,3,4,5,6,7), (1,2) >; gap:G := Group( (1,2,3,4,5,6,7), (1,2) ); sage:G = PermutationGroup(['(1,2,3,4,5,6,7)', '(1,2)'])
Matrix group:	$\left\langle \left(\begin{array}{rrrrrr} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array}\right), \left(\begin{array}{rrrrrr} 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ -1 & -1 & -1 & -1 & -1 & -1 \end{array}\right) \right\rangle \subseteq \GL_{6}(\Z)$
comment:Define the group as a matrix group with coefficients in Z magma:G := MatrixGroup< 6, Integers() \| [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, -1, -1, -1, -1, -1, -1]] >; gap:G := Group([[[1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1]], [[0, 0, 0, 0, 0, 1], [0, 0, 1, 0, 0, 0], [1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0], [-1, -1, -1, -1, -1, -1]]]); sage:MS = MatrixSpace(Integers(), 6, 6) G = MatrixGroup([MS([[1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1]]), MS([[0, 0, 0, 0, 0, 1], [0, 0, 1, 0, 0, 0], [1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0], [-1, -1, -1, -1, -1, -1]])])
Transitive group:	7T7	14T46	21T38	30T565	all 9
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$A_7$ $\,\rtimes\,$ $C_2$				more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Aut. group:	$\Aut(A_7)$	$\Aut(C_2.A_7)$	$\Aut(C_2\times A_7)$	$\Aut(C_3.A_7)$	all 8

Elements of the group are displayed as permutations of degree 7.

Homology

Abelianization:	$C_{2} $	comment:The abelianization of the group magma:quo< G \| CommutatorSubgroup(G) >; gap:FactorGroup(G, DerivedSubgroup(G)); sage:G.quotient(G.commutator())
Schur multiplier:	$C_{2}$	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 11300 subgroups in 96 conjugacy classes, 3 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $S_7$	comment:Center of the group magma:Center(G); gap:Center(G); sage:G.center() sage_gap:G.Center()
Commutator:	$G' \simeq$ $A_7$	$G/G' \simeq$ $C_2$	comment:Commutator subgroup of the group G magma:CommutatorSubgroup(G); gap:DerivedSubgroup(G); sage:G.commutator() sage_gap:G.DerivedSubgroup()
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $S_7$	comment:Frattini subgroup of the group G magma:FrattiniSubgroup(G); gap:FrattiniSubgroup(G); sage:G.frattini_subgroup() sage_gap:G.FrattiniSubgroup()
Fitting:	$\operatorname{Fit} \simeq$ $C_1$	$G/\operatorname{Fit} \simeq$ $S_7$	comment:Fitting subgroup of the group G magma:FittingSubgroup(G); gap:FittingSubgroup(G); sage:G.fitting_subgroup() sage_gap:G.FittingSubgroup()
Radical:	$R \simeq$ $C_1$	$G/R \simeq$ $S_7$	comment:Radical of the group G magma:Radical(G); gap:SolvableRadical(G); sage_gap:G.SolvableRadical()
Socle:	$\operatorname{soc} \simeq$ $A_7$	$G/\operatorname{soc} \simeq$ $C_2$	comment:Socle of the group G magma:Socle(G); gap:Socle(G); sage:G.socle() sage_gap:G.Socle()
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7$

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.

See a full page version of the diagram

Subgroup information

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

Series

Derived series

$S_7$

$\rhd$

$A_7$

comment:Derived series of the group GF

magma:DerivedSeries(G);

gap:DerivedSeriesOfGroup(G);

sage:G.derived_series()

sage_gap:G.DerivedSeriesOfGroup()

Chief series

$S_7$

$\rhd$

$A_7$

$\rhd$

$C_1$

comment:Chief series of the group G

magma:ChiefSeries(G);

gap:ChiefSeries(G);

sage_gap:G.ChiefSeries()

Lower central series

$S_7$

$\rhd$

$A_7$

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 20 larger groups in the database.

This group is a maximal quotient of 28 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

Every character has rational values, so the complex character table is the same as the rational character table below.

Rational character table

		1A	2A	2B	2C	3A	3B	4A	4B	5A	6A	6B	6C	7A	10A	12A
	Size	1	21	105	105	70	280	210	630	504	210	420	840	720	504	420
	2 P	1A	1A	1A	1A	3A	3B	2B	2B	5A	3A	3A	3B	7A	5A	6A
	3 P	1A	2A	2B	2C	1A	1A	4A	4B	5A	2B	2A	2C	7A	10A	4A
	5 P	1A	2A	2B	2C	3A	3B	4A	4B	1A	6A	6B	6C	7A	2A	12A
	7 P	1A	2A	2B	2C	3A	3B	4A	4B	5A	6A	6B	6C	1A	10A	12A
5040.w.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
5040.w.1b		$1$	$- 1$	$1$	$- 1$	$1$	$1$	$- 1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$- 1$	$- 1$
5040.w.6a		$6$	$4$	$2$	$0$	$3$	$0$	$2$	$0$	$1$	$- 1$	$1$	$0$	$- 1$	$- 1$	$- 1$
5040.w.6b		$6$	$- 4$	$2$	$0$	$3$	$0$	$- 2$	$0$	$1$	$- 1$	$- 1$	$0$	$- 1$	$1$	$1$
5040.w.14a		$14$	$6$	$2$	$2$	$2$	$- 1$	$0$	$0$	$- 1$	$2$	$0$	$- 1$	$0$	$1$	$0$
5040.w.14b		$14$	$- 4$	$2$	$0$	$- 1$	$2$	$2$	$0$	$- 1$	$- 1$	$- 1$	$0$	$0$	$1$	$- 1$
5040.w.14c		$14$	$4$	$2$	$0$	$- 1$	$2$	$- 2$	$0$	$- 1$	$- 1$	$1$	$0$	$0$	$- 1$	$1$
5040.w.14d		$14$	$- 6$	$2$	$- 2$	$2$	$- 1$	$0$	$0$	$- 1$	$2$	$0$	$1$	$0$	$- 1$	$0$
5040.w.15a		$15$	$- 5$	$- 1$	$3$	$3$	$0$	$- 1$	$- 1$	$0$	$- 1$	$1$	$0$	$1$	$0$	$- 1$
5040.w.15b		$15$	$5$	$- 1$	$- 3$	$3$	$0$	$1$	$- 1$	$0$	$- 1$	$- 1$	$0$	$1$	$0$	$1$
5040.w.20a		$20$	$0$	$- 4$	$0$	$2$	$2$	$0$	$0$	$0$	$2$	$0$	$0$	$- 1$	$0$	$0$
5040.w.21a		$21$	$- 1$	$1$	$3$	$- 3$	$0$	$1$	$- 1$	$1$	$1$	$- 1$	$0$	$0$	$- 1$	$1$
5040.w.21b		$21$	$1$	$1$	$- 3$	$- 3$	$0$	$- 1$	$- 1$	$1$	$1$	$1$	$0$	$0$	$1$	$- 1$
5040.w.35a		$35$	$5$	$- 1$	$1$	$- 1$	$- 1$	$- 1$	$1$	$0$	$- 1$	$- 1$	$1$	$0$	$0$	$- 1$
5040.w.35b		$35$	$- 5$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$1$	$0$	$- 1$	$1$	$- 1$	$0$	$0$	$1$

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Classical	Maass
Hilbert	Bianchi

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Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Label	5040.w
Order	\( 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \)
Exponent	\( 2^{2} \cdot 3 \cdot 5 \cdot 7 \)

Nilpotent	no
Solvable	no
$\card{G^{\mathrm{ab}}}$	\( 2 \)
$\card{Z(G)}$	\( 1 \)
$\card{\Aut(G)}$	\( 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \)
$\card{\mathrm{Out}(G)}$	\( 1 \)
Perm deg.	$7$
Trans deg.	$7$
Rank	$2$

Introduction

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Fields

Representations

Groups

Database

Properties

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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.