Properties

Label 14T35
14T35 1 3 1->3 9 1->9 2 4 2->4 2->4 2->9 5 3->5 13 3->13 6 4->6 8 4->8 5->3 7 5->7 6->8 12 6->12 7->9 14 7->14 8->2 10 8->10 11 9->11 9->11 10->6 10->12 11->1 11->13 12->10 12->14 13->1 13->5 14->2
Degree $14$
Order $1344$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $C_2^3:F_8:C_3$

Related objects

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Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(14, 35);
 
Copy content sage:G = TransitiveGroup(14, 35)
 
Copy content oscar:G = transitive_group(14, 35)
 
Copy content gap:G := TransitiveGroup(14, 35);
 

Group invariants

Abstract group:  $C_2^3:F_8:C_3$
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:  $1344=2^{6} \cdot 3 \cdot 7$
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 $n$:  $14$
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 $t$:  $35$
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:   $[2^{6}]F_{21}(7)$
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:  no
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);
 
$\card{\Aut(F/K)}$:  $2$
Copy content comment:Order of the centralizer of G in S_n
 
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Copy content sage:SymmetricGroup(14).centralizer(G).order()
 
Copy content oscar:order(centralizer(symmetric_group(14), G)[1])
 
Copy content gap:Order(Centralizer(SymmetricGroup(14), G));
 
Generators:  $(1,9,11)(2,4,8)(3,13,5)(6,12,10)$, $(2,9)(7,14)$, $(1,3,5,7,9,11,13)(2,4,6,8,10,12,14)$
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)
$3$:  $C_3$
$21$:  $C_7:C_3$
$168$:  $C_2^3:(C_7: C_3)$ x 2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 7: $C_7:C_3$

Low degree siblings

28T154, 28T155 x 2, 28T157, 42T202, 42T204, 42T205

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{14}$ $1$ $1$ $0$ $()$
2A $2^{4},1^{6}$ $7$ $2$ $4$ $( 1, 8)( 3,10)( 4,11)( 5,12)$
2B $2^{6},1^{2}$ $7$ $2$ $6$ $( 1, 8)( 2, 9)( 3,10)( 4,11)( 6,13)( 7,14)$
2C $2^{4},1^{6}$ $7$ $2$ $4$ $( 1, 8)( 2, 9)( 4,11)( 7,14)$
2D $2^{2},1^{10}$ $21$ $2$ $2$ $( 1, 8)( 6,13)$
2E $2^{4},1^{6}$ $21$ $2$ $4$ $( 1, 8)( 3,10)( 5,12)( 6,13)$
3A1 $3^{4},1^{2}$ $112$ $3$ $8$ $( 1, 4,10)( 2, 6,14)( 3, 8,11)( 7, 9,13)$
3A-1 $3^{4},1^{2}$ $112$ $3$ $8$ $( 1,10, 4)( 2,14, 6)( 3,11, 8)( 7,13, 9)$
6A1 $6,3^{2},2$ $112$ $6$ $10$ $( 1, 3, 4, 8,10,11)( 2,14, 6)( 5,12)( 7,13, 9)$
6A-1 $6,3^{2},2$ $112$ $6$ $10$ $( 1,11,10, 8, 4, 3)( 2, 6,14)( 5,12)( 7, 9,13)$
6B1 $6^{2},1^{2}$ $112$ $6$ $10$ $( 1, 3,11, 8,10, 4)( 2,14,13, 9, 7, 6)$
6B-1 $6^{2},1^{2}$ $112$ $6$ $10$ $( 1, 4,10, 8,11, 3)( 2, 6, 7, 9,13,14)$
6C1 $6,3^{2},2$ $112$ $6$ $10$ $( 1,11, 9, 8, 4, 2)( 3,12,13)( 5, 6,10)( 7,14)$
6C-1 $6,3^{2},2$ $112$ $6$ $10$ $( 1, 2, 4, 8, 9,11)( 3,13,12)( 5,10, 6)( 7,14)$
7A1 $7^{2}$ $192$ $7$ $12$ $( 1, 2, 3, 4, 5, 6,14)( 7, 8, 9,10,11,12,13)$
7A-1 $7^{2}$ $192$ $7$ $12$ $( 1,14, 6, 5, 4, 3, 2)( 7,13,12,11,10, 9, 8)$

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

1A 2A 2B 2C 2D 2E 3A1 3A-1 6A1 6A-1 6B1 6B-1 6C1 6C-1 7A1 7A-1
Size 1 7 7 7 21 21 112 112 112 112 112 112 112 112 192 192
2 P 1A 1A 1A 1A 1A 1A 3A-1 3A1 3A1 3A-1 3A1 3A-1 3A1 3A-1 7A1 7A-1
3 P 1A 2A 2B 2C 2D 2E 1A 1A 2A 2A 2B 2B 2C 2C 7A-1 7A1
7 P 1A 2A 2B 2C 2D 2E 3A1 3A-1 6A1 6A-1 6B1 6B-1 6C1 6C-1 1A 1A
Type
1344.11691.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1344.11691.1b1 C 1 1 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3 1 1
1344.11691.1b2 C 1 1 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1 1 1
1344.11691.3a1 C 3 3 3 3 3 3 0 0 0 0 0 0 0 0 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72
1344.11691.3a2 C 3 3 3 3 3 3 0 0 0 0 0 0 0 0 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72
1344.11691.7a R 7 −1 −1 7 −1 −1 1 1 −1 −1 −1 −1 1 1 0 0
1344.11691.7b R 7 7 −1 −1 −1 −1 1 1 1 1 −1 −1 −1 −1 0 0
1344.11691.7c R 7 −1 −5 −1 3 −1 1 1 −1 −1 1 1 −1 −1 0 0
1344.11691.7d1 C 7 −1 −1 7 −1 −1 ζ3−1 ζ3 −ζ3 −ζ3−1 −ζ3−1 −ζ3 ζ3−1 ζ3 0 0
1344.11691.7d2 C 7 −1 −1 7 −1 −1 ζ3 ζ3−1 −ζ3−1 −ζ3 −ζ3 −ζ3−1 ζ3 ζ3−1 0 0
1344.11691.7e1 C 7 7 −1 −1 −1 −1 ζ3−1 ζ3 ζ3 ζ3−1 −ζ3−1 −ζ3 −ζ3−1 −ζ3 0 0
1344.11691.7e2 C 7 7 −1 −1 −1 −1 ζ3 ζ3−1 ζ3−1 ζ3 −ζ3 −ζ3−1 −ζ3 −ζ3−1 0 0
1344.11691.7f1 C 7 −1 −5 −1 3 −1 ζ3−1 ζ3 −ζ3 −ζ3−1 ζ3−1 ζ3 −ζ3−1 −ζ3 0 0
1344.11691.7f2 C 7 −1 −5 −1 3 −1 ζ3 ζ3−1 −ζ3−1 −ζ3 ζ3 ζ3−1 −ζ3 −ζ3−1 0 0
1344.11691.21a R 21 −3 9 −3 1 −3 0 0 0 0 0 0 0 0 0 0
1344.11691.21b R 21 −3 −3 −3 −3 5 0 0 0 0 0 0 0 0 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

Data not computed