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Properties

Label 11T3
11T3 1 2 1->2 3 1->3 2->3 6 2->6 4 3->4 9 3->9 4->1 5 4->5 5->4 5->6 7 6->7 6->7 8 7->8 10 7->10 8->2 8->9 9->5 9->10 10->8 11 10->11 11->1
Degree $11$
Order $55$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive yes
$p$-group no
Group: $C_{11}:C_5$

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Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(11, 3);
 
Copy content sage:G = TransitiveGroup(11, 3)
 
Copy content oscar:G = transitive_group(11, 3)
 
Copy content gap:G := TransitiveGroup(11, 3);
 

Group invariants

Abstract group:  $C_{11}:C_5$
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:  $55=5 \cdot 11$
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 $t$:  $3$
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:   $F_{55}(11)=11:5$
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:  1
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:  $(1,3,9,5,4)(2,6,7,10,8)$, $(1,2,3,4,5,6,7,8,9,10,11)$
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)
$5$:  $C_5$

Resolvents shown for degrees $\leq 47$

Subfields

Prime degree - none

Low degree siblings

There are no siblings with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

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

Malle's constant $a(G)$:     $1/8$

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 5A1 5A-1 5A2 5A-2 11A1 11A-1
Size 1 11 11 11 11 5 5
5 P 1A 5A2 5A-2 5A-1 5A1 11A-1 11A1
11 P 1A 1A 1A 1A 1A 11A1 11A-1
Type
55.1.1a R 1 1 1 1 1 1 1
55.1.1b1 C 1 ζ5−2 ζ52 ζ5 ζ5−1 1 1
55.1.1b2 C 1 ζ52 ζ5−2 ζ5−1 ζ5 1 1
55.1.1b3 C 1 ζ5−1 ζ5 ζ5−2 ζ52 1 1
55.1.1b4 C 1 ζ5 ζ5−1 ζ52 ζ5−2 1 1
55.1.5a1 C 5 0 0 0 0 −ζ11−2−1−ζ11−ζ113−ζ114−ζ115 ζ11−2+ζ11+ζ113+ζ114+ζ115
55.1.5a2 C 5 0 0 0 0 ζ11−2+ζ11+ζ113+ζ114+ζ115 −ζ11−2−1−ζ11−ζ113−ζ114−ζ115

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

$f_{ 1 } =$ $1024 x^{11} - 2816 \left(s^{2}+11 t^{2}\right) x^{9} + 2816 \left(s^{2}+11 t^{2}\right)^{2} x^{7} - 1232 \left(s^{2}+11 t^{2}\right)^{3} x^{5} + 220 \left(s^{2}+11 t^{2}\right)^{4} x^{3} - 11 \left(s^{2}+11 t^{2}\right)^{5} x - s \left(s^{2}+11 t^{2}\right)^{5}$ Copy content Toggle raw display