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Properties

Label 14641.15
Order \( 11^{4} \)
Exponent \( 11 \)
Abelian yes
$\card{\Aut(G)}$ \( 2^{9} \cdot 3^{2} \cdot 5^{4} \cdot 7 \cdot 11^{6} \cdot 19 \cdot 61 \)
Perm deg. $44$
Trans deg. not computed
Rank $4$

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Show commands: Gap / Magma / SageMath

Copy content magma:G := SmallGroup(14641, 15);
 
Copy content gap:G := SmallGroup(14641, 15);
 
Copy content sage_gap:G = libgap.SmallGroup(14641, 15)
 
Copy content comment:Define the group as a permutation group
 
Copy content sage:G = PermutationGroup(['(1,11,10,9,8,7,6,5,4,3,2)', '(12,22,21,20,19,18,17,16,15,14,13)', '(23,33,32,31,30,29,28,27,26,25,24)', '(34,44,43,42,41,40,39,38,37,36,35)'])
 

Group information

Description:$C_{11}^4$
Order: \(14641\)\(\medspace = 11^{4} \)
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: \(11\)
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 \(41393302251840000\)\(\medspace = 2^{9} \cdot 3^{2} \cdot 5^{4} \cdot 7 \cdot 11^{6} \cdot 19 \cdot 61 \)
Copy content comment:Automorphism group
 
Copy content gap:AutomorphismGroup(G);
 
Copy content magma:AutomorphismGroup(G);
 
Copy content sage_gap:G.AutomorphismGroup()
 
Composition factors:$C_{11}$ x 4
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()
 
Nilpotency class:$1$
Copy content comment:Nilpotency class of the group
 
Copy content magma:NilpotencyClass(G);
 
Copy content gap:NilpotencyClassOfGroup(G);
 
Copy content sage_gap:G.NilpotencyClassOfGroup()
 
Derived length:$1$
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 abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group) and a $p$-group (hence elementary and hyperelementary).

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 11
Elements 1 14640 14641
Conjugacy classes   1 14640 14641
Divisions 1 1464 1465
Autjugacy classes 1 1 2

Minimal presentations

Permutation degree:$44$
Transitive degree:not computed
Rank: $4$
Inequivalent generating quadruples: not computed

Minimal degrees of faithful linear representations

Over $\mathbb{C}$ Over $\mathbb{R}$ Over $\mathbb{Q}$
Irreducible none not computed none
Arbitrary 4 not computed not computed

Constructions

Show commands: Gap / Magma / SageMath


Presentation:Abelian group $\langle a, b, c, d \mid a^{11}=b^{11}=c^{11}=d^{11}=1 \rangle$ Copy content Toggle raw display
Copy content comment:Define the group with the given generators and relations
 
Copy content magma:G := PCGroup([4, -11, 11, 11, 11]); a,b,c,d := Explode([G.1, G.2, G.3, G.4]); AssignNames(~G, ["a", "b", "c", "d"]);
 
Copy content gap:G := PcGroupCode(0,14641); a := G.1; b := G.2; c := G.3; d := G.4;
 
Copy content sage:# This uses Sage's interface to GAP, as Sage (currently) has no native support for PC groups G = gap.new('PcGroupCode(0,14641)'); a = G.1; b = G.2; c = G.3; d = G.4;
 
Copy content sage_gap:# This uses Sage's interface to GAP, as Sage (currently) has no native support for PC groups G = gap.new('PcGroupCode(0,14641)'); a = G.1; b = G.2; c = G.3; d = G.4;
 
Permutation group:Degree $44$ $\langle(1,11,10,9,8,7,6,5,4,3,2), (12,22,21,20,19,18,17,16,15,14,13), (23,33,32,31,30,29,28,27,26,25,24), (34,44,43,42,41,40,39,38,37,36,35)\rangle$ Copy content Toggle raw display
Copy content comment:Define the group as a permutation group
 
Copy content magma:G := PermutationGroup< 44 | (1,11,10,9,8,7,6,5,4,3,2), (12,22,21,20,19,18,17,16,15,14,13), (23,33,32,31,30,29,28,27,26,25,24), (34,44,43,42,41,40,39,38,37,36,35) >;
 
Copy content gap:G := Group( (1,11,10,9,8,7,6,5,4,3,2), (12,22,21,20,19,18,17,16,15,14,13), (23,33,32,31,30,29,28,27,26,25,24), (34,44,43,42,41,40,39,38,37,36,35) );
 
Copy content sage:G = PermutationGroup(['(1,11,10,9,8,7,6,5,4,3,2)', '(12,22,21,20,19,18,17,16,15,14,13)', '(23,33,32,31,30,29,28,27,26,25,24)', '(34,44,43,42,41,40,39,38,37,36,35)'])
 
Matrix group:$\left\langle \left(\begin{array}{rr} 111 & 0 \\ 0 & 111 \end{array}\right), \left(\begin{array}{rr} 56 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 34 & 110 \\ 22 & 23 \end{array}\right), \left(\begin{array}{rr} 45 & 99 \\ 22 & 12 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/121\Z)$
Copy content comment:Define the group as a matrix group with coefficients in GLZq
 
Copy content magma:G := MatrixGroup< 2, Integers(121) | [[111, 0, 0, 111], [56, 0, 0, 1], [34, 110, 22, 23], [45, 99, 22, 12]] >;
 
Copy content gap:G := Group([[[ZmodnZObj(111,121), ZmodnZObj(0,121)], [ZmodnZObj(0,121), ZmodnZObj(111,121)]],[[ZmodnZObj(56,121), ZmodnZObj(0,121)], [ZmodnZObj(0,121), ZmodnZObj(1,121)]],[[ZmodnZObj(34,121), ZmodnZObj(110,121)], [ZmodnZObj(22,121), ZmodnZObj(23,121)]],[[ZmodnZObj(45,121), ZmodnZObj(99,121)], [ZmodnZObj(22,121), ZmodnZObj(12,121)]]]);
 
Copy content sage:MS = MatrixSpace(Integers(121), 2, 2) G = MatrixGroup([MS([[111, 0], [0, 111]]), MS([[56, 0], [0, 1]]), MS([[34, 110], [22, 23]]), MS([[45, 99], [22, 12]])])
 
Direct product: $C_{11}$ ${}^4$
Semidirect product: not isomorphic to a non-trivial semidirect product
Trans. wreath product: not isomorphic to a non-trivial transitive wreath product
Possibly split product: $C_{11}^3$ . $C_{11}$ (1464) $C_{11}$ . $C_{11}^3$ (1464) more information

Elements of the group are displayed as words in the presentation generators from the presentation above.

Homology

Primary decomposition: $C_{11}^{4}$
Copy content comment:The primary decomposition of the group
 
Copy content magma:PrimaryInvariants(G);
 
Copy content gap:AbelianInvariants(G);
 
Copy content sage_gap:G.AbelianInvariants()
 
Schur multiplier: $C_{11}^{6}$
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: $0$
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 19156 normal subgroups (2 characteristic).

Characteristic subgroups are shown in this color. Normal (but not characteristic) subgroups are shown in this color.

Special subgroups

Center: a subgroup isomorphic to $C_{11}^4$
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 $C_1$
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()
 
11-Sylow subgroup: $P_{ 11 } \simeq$ $C_{11}^4$

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 6 larger groups in the database.

This group is a maximal quotient of 2 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 $14641 \times 14641$ character table is not available for this group.

Rational character table

The $1465 \times 1465$ rational character table is not available for this group.