Properties

Label 11T4
11T4 1 2 1->2 1->2 3 2->3 4 2->4 3->4 6 3->6 5 4->5 8 4->8 5->6 10 5->10 6->1 7 6->7 7->3 7->8 8->5 9 8->9 9->7 9->10 10->9 11 10->11 11->1
Degree $11$
Order $110$
Cyclic no
Abelian no
Solvable yes
Transitivity $2$
Primitive yes
$p$-group no
Group: $F_{11}$

Related objects

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Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(11, 4);
 
Copy content sage:G = TransitiveGroup(11, 4)
 
Copy content oscar:G = transitive_group(11, 4)
 
Copy content gap:G := TransitiveGroup(11, 4);
 

Group invariants

Abstract group:  $F_{11}$
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:  $110=2 \cdot 5 \cdot 11$
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:  yes
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 $t$:  $4$
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:   $F_{110}(11)=11:10$
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:  $(1,2,3,4,5,6,7,8,9,10,11)$, $(1,2,4,8,5,10,9,7,3,6)$
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

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$5$:  $C_5$
$10$:  $C_{10}$

Resolvents shown for degrees $\leq 47$

Subfields

Prime degree - none

Low degree siblings

22T4

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{11}$ $1$ $1$ $0$ $()$
2A $2^{5},1$ $11$ $2$ $5$ $( 2,11)( 3,10)( 4, 9)( 5, 8)( 6, 7)$
5A1 $5^{2},1$ $11$ $5$ $8$ $( 2, 6, 4, 5,10)( 3,11, 7, 9, 8)$
5A-1 $5^{2},1$ $11$ $5$ $8$ $( 2,10, 5, 4, 6)( 3, 8, 9, 7,11)$
5A2 $5^{2},1$ $11$ $5$ $8$ $( 2, 4,10, 6, 5)( 3, 7, 8,11, 9)$
5A-2 $5^{2},1$ $11$ $5$ $8$ $( 2, 5, 6,10, 4)( 3, 9,11, 8, 7)$
10A1 $10,1$ $11$ $10$ $9$ $( 2, 8, 6, 3, 4,11, 5, 7,10, 9)$
10A-1 $10,1$ $11$ $10$ $9$ $( 2, 9,10, 7, 5,11, 4, 3, 6, 8)$
10A3 $10,1$ $11$ $10$ $9$ $( 2, 3, 5, 9, 6,11,10, 8, 4, 7)$
10A-3 $10,1$ $11$ $10$ $9$ $( 2, 7, 4, 8,10,11, 6, 9, 5, 3)$
11A $11$ $10$ $11$ $10$ $( 1,11,10, 9, 8, 7, 6, 5, 4, 3, 2)$

Malle's constant $a(G)$:     $1/5$

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 5A1 5A-1 5A2 5A-2 10A1 10A-1 10A3 10A-3 11A
Size 1 11 11 11 11 11 11 11 11 11 10
2 P 1A 1A 5A2 5A-2 5A-1 5A1 5A1 5A-1 5A-2 5A2 11A
5 P 1A 2A 1A 1A 1A 1A 2A 2A 2A 2A 11A
11 P 1A 2A 5A1 5A-1 5A2 5A-2 10A1 10A-1 10A3 10A-3 1A
Type
110.1.1a R 1 1 1 1 1 1 1 1 1 1 1
110.1.1b R 1 −1 1 1 1 1 −1 −1 −1 −1 1
110.1.1c1 C 1 1 ζ5−2 ζ52 ζ5−1 ζ5 ζ5−2 ζ5 ζ52 ζ5−1 1
110.1.1c2 C 1 1 ζ52 ζ5−2 ζ5 ζ5−1 ζ52 ζ5−1 ζ5−2 ζ5 1
110.1.1c3 C 1 1 ζ5−1 ζ5 ζ52 ζ5−2 ζ5−1 ζ5−2 ζ5 ζ52 1
110.1.1c4 C 1 1 ζ5 ζ5−1 ζ5−2 ζ52 ζ5 ζ52 ζ5−1 ζ5−2 1
110.1.1d1 C 1 −1 ζ5−2 ζ52 ζ5−1 ζ5 −ζ5−2 −ζ5 −ζ52 −ζ5−1 1
110.1.1d2 C 1 −1 ζ52 ζ5−2 ζ5 ζ5−1 −ζ52 −ζ5−1 −ζ5−2 −ζ5 1
110.1.1d3 C 1 −1 ζ5−1 ζ5 ζ52 ζ5−2 −ζ5−1 −ζ5−2 −ζ5 −ζ52 1
110.1.1d4 C 1 −1 ζ5 ζ5−1 ζ5−2 ζ52 −ζ5 −ζ52 −ζ5−1 −ζ5−2 1
110.1.10a R 10 0 0 0 0 0 0 0 0 0 −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

Data not computed