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Properties

Label 11T6
11T6 1 2 1->2 3 1->3 2->3 6 2->6 2->6 4 3->4 9 3->9 3->9 4->1 5 4->5 4->5 5->3 5->4 5->6 7 6->7 6->7 10 6->10 7->2 8 7->8 7->10 8->2 8->9 9->4 9->5 9->10 10->7 10->8 11 10->11 11->1
Degree $11$
Order $7920$
Cyclic no
Abelian no
Solvable no
Transitivity $4$
Primitive yes
$p$-group no
Group: $M_{11}$

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, 6);
 
Copy content sage:G = TransitiveGroup(11, 6)
 
Copy content oscar:G = transitive_group(11, 6)
 
Copy content gap:G := TransitiveGroup(11, 6);
 

Group invariants

Abstract group:  $M_{11}$
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:  $7920=2^{4} \cdot 3^{2} \cdot 5 \cdot 11$
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:  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 $t$:  $6$
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:   $M(11)$
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:  4
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:  $(1,3,9,5,4)(2,6,7,10,8)$, $(2,6,10,7)(3,9,4,5)$, $(1,2,3,4,5,6,7,8,9,10,11)$
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

none

Resolvents shown for degrees $\leq 47$

Subfields

Prime degree - none

Low degree siblings

12T272, 22T22

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 $2^{4},1^{3}$ $165$ $2$ $4$ $( 1, 5)( 2, 3)( 6, 9)(10,11)$
3A $3^{3},1^{2}$ $440$ $3$ $6$ $( 1, 9,10)( 3, 8,11)( 4, 7, 5)$
4A $4^{2},1^{3}$ $990$ $4$ $6$ $( 1, 3, 5, 2)( 6,11, 9,10)$
5A $5^{2},1$ $1584$ $5$ $8$ $( 1, 8, 4, 9,10)( 2,11, 3, 5, 6)$
6A $6,3,2$ $1320$ $6$ $8$ $( 1, 3, 9, 8,10,11)( 2, 6)( 4, 5, 7)$
8A1 $8,2,1$ $990$ $8$ $8$ $( 1,11, 3, 9, 5,10, 2, 6)( 4, 7)$
8A-1 $8,2,1$ $990$ $8$ $8$ $( 1, 6, 2,10, 5, 9, 3,11)( 4, 7)$
11A1 $11$ $720$ $11$ $10$ $( 1, 9, 8,11, 3, 4,10, 6, 2, 5, 7)$
11A-1 $11$ $720$ $11$ $10$ $( 1, 7, 5, 2, 6,10, 4, 3,11, 8, 9)$

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 4A 5A 6A 8A1 8A-1 11A1 11A-1
Size 1 165 440 990 1584 1320 990 990 720 720
2 P 1A 1A 3A 2A 5A 3A 4A 4A 11A-1 11A1
3 P 1A 2A 1A 4A 5A 2A 8A1 8A-1 11A1 11A-1
5 P 1A 2A 3A 4A 1A 6A 8A-1 8A1 11A1 11A-1
11 P 1A 2A 3A 4A 5A 6A 8A1 8A-1 1A 1A
Type
7920.a.1a R 1 1 1 1 1 1 1 1 1 1
7920.a.10a R 10 2 1 2 0 −1 0 0 −1 −1
7920.a.10b1 C 10 −2 1 0 0 1 −ζ8−ζ83 ζ8+ζ83 −1 −1
7920.a.10b2 C 10 −2 1 0 0 1 ζ8+ζ83 −ζ8−ζ83 −1 −1
7920.a.11a R 11 3 2 −1 1 0 −1 −1 0 0
7920.a.16a1 C 16 0 −2 0 1 0 0 0 −ζ11−2−1−ζ11−ζ113−ζ114−ζ115 ζ11−2+ζ11+ζ113+ζ114+ζ115
7920.a.16a2 C 16 0 −2 0 1 0 0 0 ζ11−2+ζ11+ζ113+ζ114+ζ115 −ζ11−2−1−ζ11−ζ113−ζ114−ζ115
7920.a.44a R 44 4 −1 0 −1 1 0 0 0 0
7920.a.45a R 45 −3 0 1 0 0 −1 −1 1 1
7920.a.55a R 55 −1 1 −1 0 −1 1 1 0 0

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

$f_{ 1 } =$ $\left(4 t^{2} - 4 t + 1\right) x^{11} + \left(4 t^{3} + 172 t^{2} - 175 t + 44\right) x^{10} + \left(4 t^{4} + 172 t^{3} + 2841 t^{2} - 2972 t + 754\right) x^{9} + \left(4 t^{5} + 172 t^{4} + 2841 t^{3} + 21268 t^{2} - 23486 t + 6060\right) x^{8} + \left(4 t^{6} + 172 t^{5} + 2841 t^{4} + 21268 t^{3} + 51994 t^{2} - 69420 t + 18870\right) x^{7} + \left(4 t^{7} + 172 t^{6} + 2841 t^{5} + 21268 t^{4} + 51994 t^{3} - 182844 t^{2} + 132294 t - 28356\right) x^{6} + \left(4 t^{8} + 172 t^{7} + 2841 t^{6} + 21268 t^{5} + 51994 t^{4} - 182844 t^{3} - 956442 t^{2} + 1060380 t - 272184\right) x^{5} + \left(4 t^{9} + 172 t^{8} + 2841 t^{7} + 21268 t^{6} + 51994 t^{5} - 182844 t^{4} - 956442 t^{3} + 828924 t^{2} - 40728 t - 57864\right) x^{4} + \left(4 t^{10} + 172 t^{9} + 2841 t^{8} + 21268 t^{7} + 51994 t^{6} - 182844 t^{5} - 956442 t^{4} + 828924 t^{3} + 6257052 t^{2} - 6355644 t + 1574445\right) x^{3} + \left(4 t^{11} + 172 t^{10} + 2841 t^{9} + 21268 t^{8} + 51994 t^{7} - 182844 t^{6} - 956442 t^{5} + 828924 t^{4} + 6257052 t^{3} - 6727484 t^{2} + 1946285 t - 92960\right) x^{2} + \left(-4 t^{11} - 175 t^{10} - 2972 t^{9} - 23486 t^{8} - 69420 t^{7} + 132294 t^{6} + 1060380 t^{5} - 40728 t^{4} - 6355644 t^{3} + 1946285 t^{2} - 101120 t + 2060\right) x + \left(t^{11} + 44 t^{10} + 754 t^{9} + 6060 t^{8} + 18870 t^{7} - 28356 t^{6} - 272184 t^{5} - 57864 t^{4} + 1574445 t^{3} - 92960 t^{2} + 2060 t - 20\right)$ Copy content Toggle raw display