Properties

Label 11T7
11T7 1 2 1->2 3 2->3 3->1 4 3->4 5 4->5 6 5->6 7 6->7 8 7->8 9 8->9 10 9->10 11 10->11 11->3
Degree $11$
Order $19958400$
Cyclic no
Abelian no
Solvable no
Transitivity $9$
Primitive yes
$p$-group no
Group: $A_{11}$

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, 7);
 
Copy content sage:G = TransitiveGroup(11, 7)
 
Copy content oscar:G = transitive_group(11, 7)
 
Copy content gap:G := TransitiveGroup(11, 7);
 

Group invariants

Abstract group:  $A_{11}$
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:  $19958400=2^{7} \cdot 3^{4} \cdot 5^{2} \cdot 7 \cdot 11$
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 $t$:  $7$
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:   $A11$
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:  9
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:  $(1,2,3)$, $(3,4,5,6,7,8,9,10,11)$
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

none

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 $1^{11}$ $1$ $1$ $0$ $()$
2A $2^{2},1^{7}$ $990$ $2$ $2$ $( 3,10)( 4, 8)$
2B $2^{4},1^{3}$ $17325$ $2$ $4$ $( 1, 7)( 2, 5)( 4,11)( 6, 8)$
3A $3,1^{8}$ $330$ $3$ $2$ $( 3,10, 9)$
3B $3^{2},1^{5}$ $18480$ $3$ $4$ $(1,2,6)(3,8,9)$
3C $3^{3},1^{2}$ $123200$ $3$ $6$ $( 1, 9, 2)( 3, 5, 8)( 4,10,11)$
4A $4,2,1^{5}$ $41580$ $4$ $4$ $(1,4,9,8)(2,5)$
4B $4^{2},1^{3}$ $207900$ $4$ $6$ $( 1, 6, 7, 8)( 2, 4, 5,11)$
4C $4,2^{3},1$ $207900$ $4$ $6$ $( 1, 7)( 2,11)( 3, 8,10, 4)( 5, 9)$
5A $5,1^{6}$ $11088$ $5$ $4$ $( 2, 8, 7,11, 9)$
5B $5^{2},1$ $798336$ $5$ $8$ $( 1, 3, 8, 7, 5)( 2,11, 4, 6,10)$
6A $3,2^{4}$ $34650$ $6$ $6$ $( 1, 7)( 2, 5)( 3, 9,10)( 4,11)( 6, 8)$
6B $3,2^{2},1^{4}$ $69300$ $6$ $4$ $( 1, 9)( 3,11,10)( 4, 8)$
6C $3^{2},2^{2},1$ $277200$ $6$ $6$ $( 1, 2, 5)( 3,10)( 4, 8)( 7,11, 9)$
6D $6,2,1^{3}$ $554400$ $6$ $6$ $( 1, 3, 2, 8, 6, 9)( 7,11)$
6E $6,3,2$ $1108800$ $6$ $8$ $( 1, 9, 8, 4, 7, 2)( 3,11)( 5,10, 6)$
7A $7,1^{4}$ $237600$ $7$ $6$ $( 1,11, 8, 4, 6,10, 5)$
8A $8,2,1$ $2494800$ $8$ $8$ $( 1,10, 9,11, 8, 4, 5, 7)( 3, 6)$
9A $9,1^{2}$ $2217600$ $9$ $8$ $( 1,11, 5, 9, 4, 8, 2,10, 3)$
10A $5,2^{2},1^{2}$ $498960$ $10$ $6$ $( 2,11, 8, 9, 7)( 3, 4)( 6,10)$
11A1 $11$ $1814400$ $11$ $10$ $( 1, 3, 8, 4, 9, 5,10, 6,11, 7, 2)$
11A-1 $11$ $1814400$ $11$ $10$ $( 1, 2, 7,11, 6,10, 5, 9, 4, 8, 3)$
12A $4^{2},3$ $415800$ $12$ $8$ $( 1, 8, 7, 6)( 2,11, 5, 4)( 3,10, 9)$
12B $4,3,2,1^{2}$ $831600$ $12$ $6$ $( 1, 8, 9, 4)( 2, 5)( 3,10,11)$
12C $6,4,1$ $1663200$ $12$ $8$ $( 1, 9, 2, 7, 5,11)( 3, 4,10, 8)$
14A $7,2^{2}$ $712800$ $14$ $8$ $( 1, 8, 6, 7, 3, 5, 2)( 4,11)( 9,10)$
15A $5,3^{2}$ $443520$ $15$ $8$ $( 1, 4, 8)( 2, 3, 6, 7,11)( 5,10, 9)$
15B $5,3,1^{3}$ $443520$ $15$ $6$ $( 1, 3, 7)( 2,11, 6, 5, 4)$
20A $5,4,2$ $997920$ $20$ $8$ $( 1, 5)( 2, 9,11, 7, 8)( 3,10, 4, 6)$
21A1 $7,3,1$ $950400$ $21$ $8$ $( 1,10, 4,11, 5, 6, 8)( 2, 9, 7)$
21A2 $7,3,1$ $950400$ $21$ $8$ $( 1, 4, 5, 8,10,11, 6)( 2, 7, 9)$

Malle's constant $a(G)$:     $1/2$

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

31 x 31 character table

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

$f_{ 1 } =$ $x^{11} + \left(-121 t^{2} - 11\right) x + \left(110 t^{2} + 10\right)$ Copy content Toggle raw display