Properties

Label 121...000.a
Order \( 2^{17} \cdot 3^{8} \cdot 5^{4} \cdot 7^{2} \cdot 11 \cdot 13 \cdot 17 \cdot 19 \)
Exponent \( 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 \cdot 13 \cdot 17 \cdot 19 \)
Simple yes
$\card{G^{\mathrm{ab}}}$ \( 1 \)
$\card{Z(G)}$ 1
$\card{\Aut(G)}$ \( 2^{18} \cdot 3^{8} \cdot 5^{4} \cdot 7^{2} \cdot 11 \cdot 13 \cdot 17 \cdot 19 \)
$\card{\mathrm{Out}(G)}$ \( 2 \)
Perm deg. $20$
Trans deg. $20$
Rank $2$

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Show commands: Gap / Magma / SageMath

Copy content magma:G := AlternatingGroup(20);
 
Copy content gap:G := AlternatingGroup(20);
 
Copy content sage:G = AlternatingGroup(20)
 
Copy content comment:Define the group as a permutation group
 

Group information

Description:$A_{20}$
Order: \(1216451004088320000\)\(\medspace = 2^{17} \cdot 3^{8} \cdot 5^{4} \cdot 7^{2} \cdot 11 \cdot 13 \cdot 17 \cdot 19 \)
Copy content comment:Order of the group
 
Copy content magma:Order(G);
 
Copy content gap:Order(G);
 
Copy content sage:G.order()
 
Copy content sage_gap:G.Order()
 
Exponent: \(232792560\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 \cdot 13 \cdot 17 \cdot 19 \)
Copy content comment:Exponent of the group
 
Copy content magma:Exponent(G);
 
Copy content gap:Exponent(G);
 
Copy content sage:G.exponent()
 
Copy content sage_gap:G.Exponent()
 
Automorphism group:Group of order \(2432902008176640000\)\(\medspace = 2^{18} \cdot 3^{8} \cdot 5^{4} \cdot 7^{2} \cdot 11 \cdot 13 \cdot 17 \cdot 19 \)
Copy content comment:Automorphism group
 
Copy content gap:AutomorphismGroup(G);
 
Copy content magma:AutomorphismGroup(G);
 
Copy content sage_gap:G.AutomorphismGroup()
 
Composition factors:$A_{20}$
Copy content comment:Composition factors of the group
 
Copy content magma:CompositionFactors(G);
 
Copy content gap:CompositionSeries(G);
 
Copy content sage:G.composition_series()
 
Copy content sage_gap:G.CompositionSeries()
 
Derived length:$0$
Copy content comment:Derived length of the group
 
Copy content magma:DerivedLength(G);
 
Copy content gap:DerivedLength(G);
 
Copy content sage_gap:G.DerivedLength()
 

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

Copy content comment:Determine if the group G is abelian
 
Copy content magma:IsAbelian(G);
 
Copy content gap:IsAbelian(G);
 
Copy content sage:G.is_abelian()
 
Copy content sage_gap:G.IsAbelian()
 
Copy content comment:Determine if the group G is cyclic
 
Copy content magma:IsCyclic(G);
 
Copy content gap:IsCyclic(G);
 
Copy content sage:G.is_cyclic()
 
Copy content sage_gap:G.IsCyclic()
 
Copy content comment:Determine if the group G is nilpotent
 
Copy content magma:IsNilpotent(G);
 
Copy content gap:IsNilpotentGroup(G);
 
Copy content sage:G.is_nilpotent()
 
Copy content sage_gap:G.IsNilpotentGroup()
 
Copy content comment:Determine if the group G is solvable
 
Copy content magma:IsSolvable(G);
 
Copy content gap:IsSolvableGroup(G);
 
Copy content sage:G.is_solvable()
 
Copy content sage_gap:G.IsSolvableGroup()
 
Copy content comment:Determine if the group G is supersolvable
 
Copy content gap:IsSupersolvableGroup(G);
 
Copy content sage:G.is_supersolvable()
 
Copy content sage_gap:G.IsSupersolvableGroup()
 
Copy content comment:Determine if the group G is simple
 
Copy content magma:IsSimple(G);
 
Copy content gap:IsSimpleGroup(G);
 
Copy content sage_gap:G.IsSimpleGroup()
 

Group statistics

Copy content comment:Compute statistics for the group G
 
Copy content 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;
 
Copy content 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");
 
Copy content 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 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 24 26 28 30 33 35 36 39 40 42 44 45 51 52 55 56 60 63 65 66 70 72 77 78 84 90 91 99 105 110 120 126 132 140 165 168 180 210
Elements 1 11798364735 3044269834280 147223414987200 189239120970624 4300403589581400 34479959558400 6979977625420800 8470676211379200 16976210865344640 609493248000 65698234614230400 37132204032000 8807580134496000 61842055316670720 76028187755520000 23851980472320000 81549282342528000 128047474114560000 22125275344972800 3030432354892800 806359567104000 54379597079808000 3898881423360000 26584005786624000 64123306600953600 3515557054464000 6125354900121600 22808456326656000 12996271411200000 22301601741619200 69870172724352000 11519422387200000 11488703927500800 47703960944640000 23393288540160000 1843107581952000 14481559572480000 40248008530329600 14481559572480000 18714630832128000 23038844774400000 17920929970944000 16895152834560000 15798064988160000 7797762846720000 12671364625920000 3379030566912000 26735186903040000 24574767759360000 4441011602227200 5529322745856000 20274183401472000 4827186524160000 9215537909760000 6516701807616000 14744860655616000 7240779786240000 6758061133824000 8688935743488000 1216451004088320000
Conjugacy classes   1 5 6 12 4 30 2 10 5 11 1 41 1 6 13 2 1 8 2 13 6 2 15 1 6 24 3 3 6 2 5 10 2 4 2 1 1 2 15 3 1 2 3 1 1 1 6 1 2 2 3 1 3 1 1 2 2 1 1 2 324
Divisions 1 5 6 12 4 30 2 10 5 11 1 41 1 6 12 2 1 8 1 13 6 2 15 1 6 24 3 3 6 2 5 10 2 4 1 1 1 2 15 2 1 2 3 1 1 1 6 1 1 1 3 1 3 1 1 2 1 1 1 2 317
Autjugacy classes 1 5 6 12 4 30 2 10 5 11 1 41 1 6 12 2 1 8 1 13 6 2 15 1 6 24 3 3 6 2 5 10 2 4 1 1 1 2 15 2 1 2 3 1 1 1 6 1 1 1 3 1 3 1 1 2 1 1 1 2 317

Minimal presentations

Permutation degree:$20$
Transitive degree:$20$
Rank: $2$
Inequivalent generating pairs: not computed

Minimal degrees of faithful linear representations

Over $\mathbb{C}$ Over $\mathbb{R}$ Over $\mathbb{Q}$
Irreducible 19 not computed not computed
Arbitrary not computed not computed not computed

Constructions

Show commands: Gap / Magma / SageMath


Permutation group:Degree $20$ $\langle(1,2)(3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20), (1,2,3)\rangle$ Copy content Toggle raw display
Copy content comment:Define the group as a permutation group
 
Copy content magma:G := PermutationGroup< 20 | (1,2)(3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20), (1,2,3) >;
 
Copy content gap:G := Group( (1,2)(3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20), (1,2,3) );
 
Copy content sage:G = PermutationGroup(['(1,2)(3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20)', '(1,2,3)'])
 
Transitive group: 20T1116 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

Elements of the group are displayed as permutations of degree 20.

Homology

Abelianization: $C_1 $
Copy content comment:The abelianization of the group
 
Copy content magma:quo< G | CommutatorSubgroup(G) >;
 
Copy content gap:FactorGroup(G, DerivedSubgroup(G));
 
Copy content sage:G.quotient(G.commutator())
 
Schur multiplier: not computed
Copy content comment:The Schur multiplier of the group
 
Copy content gap:AbelianInvariantsMultiplier(G);
 
Copy content sage:G.homology(2)
 
Copy content sage_gap:G.AbelianInvariantsMultiplier()
 
Commutator length: $1$
Copy content comment:The commutator length of the group
 
Copy content gap:CommutatorLength(G);
 
Copy content sage_gap:G.CommutatorLength()
 

Subgroups

Copy content comment:List of subgroups of the group
 
Copy content magma:Subgroups(G);
 
Copy content gap:AllSubgroups(G);
 
Copy content sage:G.subgroups()
 
Copy content sage_gap:G.AllSubgroups()
 

There are 2 normal subgroups, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center: a subgroup isomorphic to $C_1$
Copy content comment:Center of the group
 
Copy content magma:Center(G);
 
Copy content gap:Center(G);
 
Copy content sage:G.center()
 
Copy content sage_gap:G.Center()
 
Commutator: a subgroup isomorphic to $A_{20}$
Copy content comment:Commutator subgroup of the group G
 
Copy content magma:CommutatorSubgroup(G);
 
Copy content gap:DerivedSubgroup(G);
 
Copy content sage:G.commutator()
 
Copy content sage_gap:G.DerivedSubgroup()
 
Frattini: a subgroup isomorphic to $C_1$
Copy content comment:Frattini subgroup of the group G
 
Copy content magma:FrattiniSubgroup(G);
 
Copy content gap:FrattiniSubgroup(G);
 
Copy content sage:G.frattini_subgroup()
 
Copy content sage_gap:G.FrattiniSubgroup()
 
Fitting: not computed
Copy content comment:Fitting subgroup of the group G
 
Copy content magma:FittingSubgroup(G);
 
Copy content gap:FittingSubgroup(G);
 
Copy content sage:G.fitting_subgroup()
 
Copy content sage_gap:G.FittingSubgroup()
 
Radical: not computed
Copy content comment:Radical of the group G
 
Copy content magma:Radical(G);
 
Copy content gap:SolvableRadical(G);
 
Copy content sage_gap:G.SolvableRadical()
 
Socle: not computed
Copy content comment:Socle of the group G
 
Copy content magma:Socle(G);
 
Copy content gap:Socle(G);
 
Copy content sage:G.socle()
 
Copy content sage_gap:G.Socle()
 
2-Sylow subgroup: $P_{ 2 } \simeq$ $C_2^5.C_2^5.C_2^4.C_2^3$
3-Sylow subgroup: $P_{ 3 } \simeq$ $C_3^4.C_3^4$
5-Sylow subgroup: $P_{ 5 } \simeq$ $C_5^4$
7-Sylow subgroup: $P_{ 7 } \simeq$ $C_7^2$
11-Sylow subgroup: $P_{ 11 } \simeq$ $C_{11}$
13-Sylow subgroup: $P_{ 13 } \simeq$ $C_{13}$
17-Sylow subgroup: $P_{ 17 } \simeq$ $C_{17}$
19-Sylow subgroup: $P_{ 19 } \simeq$ $C_{19}$

Subgroup diagram and profile

Series

Derived series not computed
Copy content comment:Derived series of the group GF
 
Copy content magma:DerivedSeries(G);
 
Copy content gap:DerivedSeriesOfGroup(G);
 
Copy content sage:G.derived_series()
 
Copy content sage_gap:G.DerivedSeriesOfGroup()
 
Chief series not computed
Copy content comment:Chief series of the group G
 
Copy content magma:ChiefSeries(G);
 
Copy content gap:ChiefSeries(G);
 
Copy content sage_gap:G.ChiefSeries()
 
Lower central series not computed
Copy content comment:The lower central series of the group G
 
Copy content magma:LowerCentralSeries(G);
 
Copy content gap:LowerCentralSeriesOfGroup(G);
 
Copy content sage:G.lower_central_series()
 
Copy content sage_gap:G.LowerCentralSeriesOfGroup()
 
Upper central series not computed
Copy content comment:The upper central series of the group G
 
Copy content magma:UpperCentralSeries(G);
 
Copy content gap:UpperCentralSeriesOfGroup(G);
 
Copy content sage:G.upper_central_series()
 
Copy content sage_gap:G.UpperCentralSeriesOfGroup()
 

Supergroups

This group is a maximal subgroup of 3 larger groups in the database.

This group is a maximal quotient of 1 larger groups in the database.

Character theory

Copy content comment:Character table
 
Copy content magma:CharacterTable(G); // Output not guaranteed to exactly match the LMFDB table
 
Copy content gap:CharacterTable(G); # Output not guaranteed to exactly match the LMFDB table
 
Copy content sage:G.character_table() # Output not guaranteed to exactly match the LMFDB table
 
Copy content sage_gap:G.CharacterTable() # Output not guaranteed to exactly match the LMFDB table
 

Complex character table

The $324 \times 324$ character table is not available for this group.

Rational character table

The $317 \times 317$ rational character table is not available for this group.