Properties

Label 11T5
11T5 1 2 1->2 3 2->3 10 2->10 4 3->4 3->4 5 4->5 6 5->6 9 5->9 7 6->7 6->7 8 7->8 8->9 9->10 11 10->11 11->1
Degree $11$
Order $660$
Cyclic no
Abelian no
Solvable no
Transitivity $2$
Primitive yes
$p$-group no
Group: $\PSL(2,11)$

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Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(11, 5);
 
Copy content sage:G = TransitiveGroup(11, 5)
 
Copy content oscar:G = transitive_group(11, 5)
 
Copy content gap:G := TransitiveGroup(11, 5);
 

Group invariants

Abstract group:  $\PSL(2,11)$
Copy content comment:Abstract group ID
 
Copy content magma:IdentifyGroup(G);
 
Copy content sage:G.id()
 
Copy content oscar:small_group_identification(G)
 
Copy content gap:IdGroup(G);
 
Order:  $660=2^{2} \cdot 3 \cdot 5 \cdot 11$
Copy content comment:Order
 
Copy content magma:Order(G);
 
Copy content sage:G.order()
 
Copy content oscar:order(G)
 
Copy content gap:Order(G);
 
Cyclic:  no
Copy content comment:Determine if group is cyclic
 
Copy content magma:IsCyclic(G);
 
Copy content sage:G.is_cyclic()
 
Copy content oscar:is_cyclic(G)
 
Copy content gap:IsCyclic(G);
 
Abelian:  no
Copy content comment:Determine if group is abelian
 
Copy content magma:IsAbelian(G);
 
Copy content sage:G.is_abelian()
 
Copy content oscar:is_abelian(G)
 
Copy content gap:IsAbelian(G);
 
Solvable:  no
Copy content comment:Determine if group is solvable
 
Copy content magma:IsSolvable(G);
 
Copy content sage:G.is_solvable()
 
Copy content oscar:is_solvable(G)
 
Copy content gap:IsSolvable(G);
 
Nilpotency class:   not nilpotent
Copy content comment:Nilpotency class
 
Copy content magma:NilpotencyClass(G);
 
Copy content sage:libgap(G).NilpotencyClassOfGroup() if G.is_nilpotent() else -1
 
Copy content oscar:if is_nilpotent(G) nilpotency_class(G) end
 
Copy content gap:if IsNilpotentGroup(G) then NilpotencyClassOfGroup(G); fi;
 

Group action invariants

Degree $n$:  $11$
Copy content comment:Degree
 
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Copy content sage:G.degree()
 
Copy content oscar:degree(G)
 
Copy content gap:NrMovedPoints(G);
 
Transitive number $t$:  $5$
Copy content comment:Transitive number
 
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Copy content sage:G.transitive_number()
 
Copy content oscar:transitive_group_identification(G)[2]
 
Copy content gap:TransitiveIdentification(G);
 
CHM label:   $L(11)=PSL(2,11)(11)$
Parity:  $1$
Copy content comment:Parity
 
Copy content magma:IsEven(G);
 
Copy content sage:all(g.SignPerm() == 1 for g in libgap(G).GeneratorsOfGroup())
 
Copy content oscar:is_even(G)
 
Copy content gap:ForAll(GeneratorsOfGroup(G), g -> SignPerm(g) = 1);
 
Transitivity:  2
Primitive:  yes
Copy content comment:Determine if group is primitive
 
Copy content magma:IsPrimitive(G);
 
Copy content sage:G.is_primitive()
 
Copy content oscar:is_primitive(G)
 
Copy content gap:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $1$
Copy content comment:Order of the centralizer of G in S_n
 
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Copy content sage:SymmetricGroup(11).centralizer(G).order()
 
Copy content oscar:order(centralizer(symmetric_group(11), G)[1])
 
Copy content gap:Order(Centralizer(SymmetricGroup(11), G));
 
Generators:  $(1,2,3,4,5,6,7,8,9,10,11)$, $(2,10)(3,4)(5,9)(6,7)$
Copy content comment:Generators
 
Copy content magma:Generators(G);
 
Copy content sage:G.gens()
 
Copy content oscar:gens(G)
 
Copy content gap:GeneratorsOfGroup(G);
 

Low degree resolvents

none

Resolvents shown for degrees $\leq 47$

Subfields

Prime degree - none

Low degree siblings

11T5, 12T179

Siblings are shown with degree $\leq 47$

A number field with this Galois group has exactly one arithmetically equivalent field.

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{11}$ $1$ $1$ $0$ $()$
2A $2^{4},1^{3}$ $55$ $2$ $4$ $( 1,10)( 3, 9)( 4, 6)( 8,11)$
3A $3^{3},1^{2}$ $110$ $3$ $6$ $( 2, 7, 5)( 3,11, 6)( 4, 9, 8)$
5A1 $5^{2},1$ $132$ $5$ $8$ $( 1, 2, 7,11, 4)( 3, 6, 9, 5,10)$
5A2 $5^{2},1$ $132$ $5$ $8$ $( 1, 7, 4, 2,11)( 3, 9,10, 6, 5)$
6A $6,3,2$ $110$ $6$ $8$ $( 1,10)( 2, 5, 7)( 3, 4,11, 9, 6, 8)$
11A1 $11$ $60$ $11$ $10$ $( 1, 4, 8, 5, 3, 7, 9, 6, 2,10,11)$
11A-1 $11$ $60$ $11$ $10$ $( 1,11,10, 2, 6, 9, 7, 3, 5, 8, 4)$

Malle's constant $a(G)$:     $1/4$

Copy content comment:Conjugacy classes
 
Copy content magma:ConjugacyClasses(G);
 
Copy content sage:G.conjugacy_classes()
 
Copy content oscar:conjugacy_classes(G)
 
Copy content gap:ConjugacyClasses(G);
 

Character table

1A 2A 3A 5A1 5A2 6A 11A1 11A-1
Size 1 55 110 132 132 110 60 60
2 P 1A 1A 3A 5A2 5A1 3A 11A-1 11A1
3 P 1A 2A 1A 5A2 5A1 2A 11A1 11A-1
5 P 1A 2A 3A 1A 1A 6A 11A1 11A-1
11 P 1A 2A 3A 5A1 5A2 6A 1A 1A
Type
660.13.1a R 1 1 1 1 1 1 1 1
660.13.5a1 C 5 1 −1 0 0 1 −ζ11−2−1−ζ11−ζ113−ζ114−ζ115 ζ11−2+ζ11+ζ113+ζ114+ζ115
660.13.5a2 C 5 1 −1 0 0 1 ζ11−2+ζ11+ζ113+ζ114+ζ115 −ζ11−2−1−ζ11−ζ113−ζ114−ζ115
660.13.10a R 10 2 1 0 0 −1 −1 −1
660.13.10b R 10 −2 1 0 0 1 −1 −1
660.13.11a R 11 −1 −1 1 1 −1 0 0
660.13.12a1 R 12 0 0 ζ5−2+ζ52 ζ5−1+ζ5 0 1 1
660.13.12a2 R 12 0 0 ζ5−1+ζ5 ζ5−2+ζ52 0 1 1

Copy content comment:Character table
 
Copy content magma:CharacterTable(G);
 
Copy content sage:G.character_table()
 
Copy content oscar:character_table(G)
 
Copy content gap:CharacterTable(G);
 

Regular extensions

$f_{ 1 } =$ $x^{11} - 3 x^{10} + 7 x^{9} - 25 x^{8} + 46 x^{7} - 36 x^{6} + t x^{5} + \left(-3 t + 60\right) x^{4} + \left(3 t - 121\right) x^{3} + \left(-t + 140\right) x^{2} - 95 x + 27$ Copy content Toggle raw display