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Properties

Label 1320.13
Order \( 2^{3} \cdot 3 \cdot 5 \cdot 11 \)
Exponent \( 2^{2} \cdot 3 \cdot 5 \cdot 11 \)
Nilpotent no
Solvable no
$\card{G^{\mathrm{ab}}}$ \( 1 \)
$\card{Z(G)}$ \( 2 \)
$\card{\Aut(G)}$ \( 2^{3} \cdot 3 \cdot 5 \cdot 11 \)
$\card{\mathrm{Out}(G)}$ \( 2 \)
Perm deg. $24$
Trans deg. $24$
Rank $2$

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Show commands: Gap / Magma / SageMath

Copy content magma:G := SL(2, 11);
 
Copy content gap:G := SL(2, 11);
 
Copy content sage:G = SL(2, 11)
 
Copy content sage_gap:G = libgap.SmallGroup(1320, 13)
 
Copy content comment:Define the group as a permutation group
 

Group information

Description:$\SL(2,11)$
Order: \(1320\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 11 \)
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: \(660\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 11 \)
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:$\PGL(2,11)$, of order \(1320\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 11 \)
Copy content comment:Automorphism group
 
Copy content gap:AutomorphismGroup(G);
 
Copy content magma:AutomorphismGroup(G);
 
Copy content sage_gap:G.AutomorphismGroup()
 
Composition factors:$C_2$, $\PSL(2,11)$
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 quasisimple (hence nonsolvable and perfect).

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 10 11 12 22
Elements 1 1 110 110 264 110 264 120 220 120 1320
Conjugacy classes   1 1 1 1 2 1 2 2 2 2 15
Divisions 1 1 1 1 1 1 1 1 1 1 10
Autjugacy classes 1 1 1 1 2 1 2 1 2 1 13

Copy content comment:Compute statistics about the characters of G
 
Copy content 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);
 
Copy content 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);
 
Copy content 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)]
 
Copy content sage_gap:G.CharacterDegrees()
 

Dimension 1 5 6 10 11 12 20 24
Irr. complex chars.   1 2 2 5 1 4 0 0 15
Irr. rational chars. 1 0 0 4 1 1 1 2 10

Minimal presentations

Permutation degree:$24$
Transitive degree:$24$
Rank: $2$
Inequivalent generating pairs: $1016$

Minimal degrees of faithful linear representations

Over $\mathbb{C}$ Over $\mathbb{R}$ Over $\mathbb{Q}$
Irreducible 6 12 12
Arbitrary 6 12 12

Constructions

Show commands: Gap / Magma / SageMath


Groups of Lie type:$\SL(2,11)$, $\SU(2,11)$
Permutation group:Degree $24$ $\langle(1,3,2)(4,7,5)(6,8,11)(9,12,16)(10,13,18)(14,19,23)(15,17,21)(20,22,24) \!\cdots\! \rangle$ Copy content Toggle raw display
Copy content comment:Define the group as a permutation group
 
Copy content magma:G := PermutationGroup< 24 | (1,3,2)(4,7,5)(6,8,11)(9,12,16)(10,13,18)(14,19,23)(15,17,21)(20,22,24), (1,2,4,5)(3,6,7,9)(8,10,12,14)(11,15,16,20)(13,17,19,22)(18,21,23,24), (1,4)(2,5)(3,7)(6,9)(8,12)(10,14)(11,16)(13,19)(15,20)(17,22)(18,23)(21,24) >;
 
Copy content gap:G := Group( (1,3,2)(4,7,5)(6,8,11)(9,12,16)(10,13,18)(14,19,23)(15,17,21)(20,22,24), (1,2,4,5)(3,6,7,9)(8,10,12,14)(11,15,16,20)(13,17,19,22)(18,21,23,24), (1,4)(2,5)(3,7)(6,9)(8,12)(10,14)(11,16)(13,19)(15,20)(17,22)(18,23)(21,24) );
 
Copy content sage:G = PermutationGroup(['(1,3,2)(4,7,5)(6,8,11)(9,12,16)(10,13,18)(14,19,23)(15,17,21)(20,22,24)', '(1,2,4,5)(3,6,7,9)(8,10,12,14)(11,15,16,20)(13,17,19,22)(18,21,23,24)', '(1,4)(2,5)(3,7)(6,9)(8,12)(10,14)(11,16)(13,19)(15,20)(17,22)(18,23)(21,24)'])
 
Matrix group:$\left\langle \left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 0 \\ 1 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{11})$
Copy content comment:Define the group as a matrix group with coefficients in GLFp
 
Copy content magma:G := MatrixGroup< 2, GF(11) | [[1, 1, 0, 1], [1, 0, 1, 1]] >;
 
Copy content gap:G := Group([[[ Z(11)^0, Z(11)^0 ], [ 0*Z(11), Z(11)^0 ]], [[ Z(11)^0, 0*Z(11) ], [ Z(11)^0, Z(11)^0 ]]]);
 
Copy content sage:MS = MatrixSpace(GF(11), 2, 2) G = MatrixGroup([MS([[1, 1], [0, 1]]), MS([[1, 0], [1, 1]])])
 
Transitive group: 24T2947 more information
Direct product: not isomorphic to a non-trivial direct product
Semidirect product: not computed
Trans. wreath product: not isomorphic to a non-trivial transitive wreath product
Non-split product: $C_2$ . $\PSL(2,11)$ more information

Elements of the group are displayed as matrices in $\SL(2,11)$.

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: $C_1$
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 766 subgroups in 21 conjugacy classes, 3 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center: $Z \simeq$ $C_2$ $G/Z \simeq$ $\PSL(2,11)$
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: $G' \simeq$ $\SL(2,11)$ $G/G' \simeq$ $C_1$
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: $\Phi \simeq$ $C_2$ $G/\Phi \simeq$ $\PSL(2,11)$
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: $\operatorname{Fit} \simeq$ $C_2$ $G/\operatorname{Fit} \simeq$ $\PSL(2,11)$
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: $R \simeq$ $C_2$ $G/R \simeq$ $\PSL(2,11)$
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: $\operatorname{soc} \simeq$ $C_2$ $G/\operatorname{soc} \simeq$ $\PSL(2,11)$
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: $P_{ 2 } \simeq$ $Q_8$
3-Sylow subgroup: $P_{ 3 } \simeq$ $C_3$
5-Sylow subgroup: $P_{ 5 } \simeq$ $C_5$
11-Sylow subgroup: $P_{ 11 } \simeq$ $C_{11}$

Subgroup diagram and profile

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Subgroup information

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Series

Derived series $\SL(2,11)$
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 $\SL(2,11)$ $\rhd$ $C_2$ $\rhd$ $C_1$
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 $\SL(2,11)$
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 $C_1$ $\lhd$ $C_2$
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 12 larger groups in the database.

This group is a maximal quotient of 8 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

1A 2A 3A 4A 5A1 5A2 6A 10A1 10A3 11A1 11A-1 12A1 12A5 22A1 22A-1
Size 1 1 110 110 132 132 110 132 132 60 60 110 110 60 60
2 P 1A 1A 3A 2A 5A2 5A1 3A 5A1 5A2 11A-1 11A1 6A 6A 11A1 11A-1
3 P 1A 2A 1A 4A 5A2 5A1 2A 10A3 10A1 11A1 11A-1 4A 4A 22A1 22A-1
5 P 1A 2A 3A 4A 1A 1A 6A 2A 2A 11A1 11A-1 12A5 12A1 22A1 22A-1
11 P 1A 2A 3A 4A 5A1 5A2 6A 10A1 10A3 1A 1A 12A1 12A5 2A 2A
Type
1320.13.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1320.13.5a1 C 5 5 −1 1 0 0 −1 0 0 −ζ11−2−1−ζ11−ζ113−ζ114−ζ115 ζ11−2+ζ11+ζ113+ζ114+ζ115 1 1 ζ11−2+ζ11+ζ113+ζ114+ζ115 −ζ11−2−1−ζ11−ζ113−ζ114−ζ115
1320.13.5a2 C 5 5 −1 1 0 0 −1 0 0 ζ11−2+ζ11+ζ113+ζ114+ζ115 −ζ11−2−1−ζ11−ζ113−ζ114−ζ115 1 1 −ζ11−2−1−ζ11−ζ113−ζ114−ζ115 ζ11−2+ζ11+ζ113+ζ114+ζ115
1320.13.6a1 C 6 −6 0 0 1 1 0 −1 −1 −ζ11−2−ζ11−ζ113−ζ114−ζ115 ζ11−2+1+ζ11+ζ113+ζ114+ζ115 0 0 −ζ11−2−1−ζ11−ζ113−ζ114−ζ115 ζ11−2+ζ11+ζ113+ζ114+ζ115
1320.13.6a2 C 6 −6 0 0 1 1 0 −1 −1 ζ11−2+1+ζ11+ζ113+ζ114+ζ115 −ζ11−2−ζ11−ζ113−ζ114−ζ115 0 0 ζ11−2+ζ11+ζ113+ζ114+ζ115 −ζ11−2−1−ζ11−ζ113−ζ114−ζ115
1320.13.10a R 10 10 1 2 0 0 1 0 0 −1 −1 −1 −1 −1 −1
1320.13.10b R 10 10 1 −2 0 0 1 0 0 −1 −1 1 1 −1 −1
1320.13.10c S 10 −10 −2 0 0 0 2 0 0 −1 −1 0 0 1 1
1320.13.10d1 S 10 −10 1 0 0 0 −1 0 0 −1 −1 −ζ12−1−ζ12 ζ12−1+ζ12 1 1
1320.13.10d2 S 10 −10 1 0 0 0 −1 0 0 −1 −1 ζ12−1+ζ12 −ζ12−1−ζ12 1 1
1320.13.11a R 11 11 −1 −1 1 1 −1 1 1 0 0 −1 −1 0 0
1320.13.12a1 R 12 12 0 0 ζ5−2+ζ52 ζ5−1+ζ5 0 ζ5−1+ζ5 ζ5−2+ζ52 1 1 0 0 1 1
1320.13.12a2 R 12 12 0 0 ζ5−1+ζ5 ζ5−2+ζ52 0 ζ5−2+ζ52 ζ5−1+ζ5 1 1 0 0 1 1
1320.13.12b1 S 12 −12 0 0 ζ5−2+ζ52 ζ5−1+ζ5 0 −ζ5−1−ζ5 −ζ5−2−ζ52 1 1 0 0 −1 −1
1320.13.12b2 S 12 −12 0 0 ζ5−1+ζ5 ζ5−2+ζ52 0 −ζ5−2−ζ52 −ζ5−1−ζ5 1 1 0 0 −1 −1

Rational character table

1A 2A 3A 4A 5A 6A 10A 11A 12A 22A
Size 1 1 110 110 264 110 264 120 220 120
2 P 1A 1A 3A 2A 5A 3A 5A 11A 6A 11A
3 P 1A 2A 1A 4A 5A 2A 10A 11A 4A 22A
5 P 1A 2A 3A 4A 1A 6A 2A 11A 12A 22A
11 P 1A 2A 3A 4A 5A 6A 10A 1A 12A 2A
Schur
1320.13.1a 1 1 1 1 1 1 1 1 1 1
1320.13.5a 10 10 −2 2 0 −2 0 −1 2 −1
1320.13.6a 12 −12 0 0 2 0 −2 1 0 −1
1320.13.10a 10 10 1 2 0 1 0 −1 −1 −1
1320.13.10b 10 10 1 −2 0 1 0 −1 1 −1
1320.13.10c 2 10 −10 −2 0 0 2 0 −1 0 1
1320.13.10d 2 20 −20 2 0 0 −2 0 −2 0 2
1320.13.11a 11 11 −1 −1 1 −1 1 0 −1 0
1320.13.12a 24 24 0 0 −1 0 −1 2 0 2
1320.13.12b 2 24 −24 0 0 −1 0 1 2 0 −2