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Properties

Label 14T36
14T36 1 6 1->6 13 1->13 2 4 2->4 2->4 9 2->9 12 2->12 3 3->4 11 3->11 4->6 8 4->8 10 4->10 4->11 5 5->2 5->9 6->8 6->8 6->12 6->13 7 14 7->14 8->1 8->2 8->10 9->12 10->3 10->6 10->12 11->10 12->5 12->10 12->14 13->8 14->2
Degree $14$
Order $1764$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $C_7^2:C_3:C_{12}$

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, 36);
 
Copy content sage:G = TransitiveGroup(14, 36)
 
Copy content oscar:G = transitive_group(14, 36)
 
Copy content gap:G := TransitiveGroup(14, 36);
 

Group invariants

Abstract group:  $C_7^2:C_3:C_{12}$
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:  $1764=2^{2} \cdot 3^{2} \cdot 7^{2}$
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 $t$:  $36$
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:   $1/2[F_{42}(7)^{2}]2$
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);
 
$\card{\Aut(F/K)}$:  $1$
Copy content comment:Order of the centralizer of G in S_n
 
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Copy content sage:SymmetricGroup(14).centralizer(G).order()
 
Copy content oscar:order(centralizer(symmetric_group(14), G)[1])
 
Copy content gap:Order(Centralizer(SymmetricGroup(14), G));
 
Generators:  $(2,4,6,8,10,12,14)$, $(2,4,8)(6,12,10)$, $(1,6,13,8)(2,9,12,5)(3,4,11,10)(7,14)$, $(1,13)(2,12)(3,11)(4,10)(5,9)(6,8)$
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)
$2$:  $C_2$
$3$:  $C_3$
$4$:  $C_4$
$6$:  $S_3$, $C_6$
$12$:  $C_{12}$, $C_3 : C_4$
$18$:  $S_3\times C_3$
$36$:  $C_3\times (C_3 : C_4)$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 7: None

Low degree siblings

28T169, 42T248, 42T249, 42T250, 42T251, 42T257

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 $2^{6},1^{2}$ $49$ $2$ $6$ $( 2, 4)( 3,13)( 5,11)( 6,14)( 7, 9)( 8,12)$
3A1 $3^{2},1^{8}$ $14$ $3$ $4$ $( 3, 9, 5)( 7,11,13)$
3A-1 $3^{2},1^{8}$ $14$ $3$ $4$ $( 3, 5, 9)( 7,13,11)$
3B1 $3^{4},1^{2}$ $49$ $3$ $8$ $( 1, 5, 7)( 2,12,10)( 3,13,11)( 4, 6,14)$
3B-1 $3^{4},1^{2}$ $49$ $3$ $8$ $( 1, 7, 5)( 2,10,12)( 3,11,13)( 4,14, 6)$
3C $3^{4},1^{2}$ $98$ $3$ $8$ $( 2, 6, 8)( 3, 5, 9)( 4,14,12)( 7,13,11)$
4A1 $4^{3},2$ $147$ $4$ $10$ $( 1,10, 3, 6)( 2,13,14, 5)( 4, 7,12,11)( 8, 9)$
4A-1 $4^{3},2$ $147$ $4$ $10$ $( 1, 6, 3,10)( 2, 5,14,13)( 4,11,12, 7)( 8, 9)$
6A1 $6^{2},1^{2}$ $49$ $6$ $10$ $( 1,11, 5, 3, 7,13)( 2, 6,12,14,10, 4)$
6A-1 $6^{2},1^{2}$ $49$ $6$ $10$ $( 1,13, 7, 3, 5,11)( 2, 4,10,14,12, 6)$
6B $6^{2},1^{2}$ $98$ $6$ $10$ $( 2,12, 6, 4, 8,14)( 3, 7, 5,13, 9,11)$
6C1 $6,2^{3},1^{2}$ $98$ $6$ $8$ $( 2, 4)( 3,11, 9,13, 5, 7)( 6,14)( 8,12)$
6C-1 $6,2^{3},1^{2}$ $98$ $6$ $8$ $( 2, 4)( 3, 7, 5,13, 9,11)( 6,14)( 8,12)$
7A $7,1^{7}$ $12$ $7$ $6$ $( 1, 3, 5, 7, 9,11,13)$
7B $7^{2}$ $36$ $7$ $12$ $( 1,13,11, 9, 7, 5, 3)( 2,10, 4,12, 6,14, 8)$
12A1 $12,2$ $147$ $12$ $12$ $( 1,14,11,10, 5, 4, 3, 2, 7, 6,13,12)( 8, 9)$
12A-1 $12,2$ $147$ $12$ $12$ $( 1,12,13, 6, 7, 2, 3, 4, 5,10,11,14)( 8, 9)$
12A5 $12,2$ $147$ $12$ $12$ $( 1, 4,13,10, 7,14, 3,12, 5, 6,11, 2)( 8, 9)$
12A-5 $12,2$ $147$ $12$ $12$ $( 1, 2,11, 6, 5,12, 3,14, 7,10,13, 4)( 8, 9)$
21A1 $7,3^{2},1$ $84$ $21$ $10$ $( 1,11, 7, 3,13, 9, 5)( 4, 6,10)( 8,14,12)$
21A-1 $7,3^{2},1$ $84$ $21$ $10$ $( 1, 5, 9,13, 3, 7,11)( 4,10, 6)( 8,12,14)$

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 3A1 3A-1 3B1 3B-1 3C 4A1 4A-1 6A1 6A-1 6B 6C1 6C-1 7A 7B 12A1 12A-1 12A5 12A-5 21A1 21A-1
Size 1 49 14 14 49 49 98 147 147 49 49 98 98 98 12 36 147 147 147 147 84 84
2 P 1A 1A 3A-1 3A1 3B-1 3B1 3C 2A 2A 3B1 3B-1 3C 3A1 3A-1 7A 7B 6A1 6A-1 6A-1 6A1 21A-1 21A1
3 P 1A 2A 1A 1A 1A 1A 1A 4A-1 4A1 2A 2A 2A 2A 2A 7A 7B 4A1 4A-1 4A1 4A-1 7A 7A
7 P 1A 2A 3A1 3A-1 3B1 3B-1 3C 4A-1 4A1 6A1 6A-1 6B 6C1 6C-1 1A 1A 12A-5 12A5 12A-1 12A1 3A-1 3A1
Type
1764.133.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1764.133.1b R 1 1 1 1 1 1 1 −1 −1 1 1 1 1 1 1 1 −1 −1 −1 −1 1 1
1764.133.1c1 C 1 1 ζ3−1 ζ3 ζ3 ζ3−1 1 1 1 ζ3−1 ζ3 1 ζ3 ζ3−1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1
1764.133.1c2 C 1 1 ζ3 ζ3−1 ζ3−1 ζ3 1 1 1 ζ3 ζ3−1 1 ζ3−1 ζ3 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3
1764.133.1d1 C 1 −1 1 1 1 1 1 −i i −1 −1 −1 −1 −1 1 1 i −i i −i 1 1
1764.133.1d2 C 1 −1 1 1 1 1 1 i −i −1 −1 −1 −1 −1 1 1 −i i −i i 1 1
1764.133.1e1 C 1 1 ζ3−1 ζ3 ζ3 ζ3−1 1 −1 −1 ζ3−1 ζ3 1 ζ3 ζ3−1 1 1 −ζ3 −ζ3−1 −ζ3−1 −ζ3 ζ3 ζ3−1
1764.133.1e2 C 1 1 ζ3 ζ3−1 ζ3−1 ζ3 1 −1 −1 ζ3 ζ3−1 1 ζ3−1 ζ3 1 1 −ζ3−1 −ζ3 −ζ3 −ζ3−1 ζ3−1 ζ3
1764.133.1f1 C 1 −1 −ζ122 ζ124 ζ124 −ζ122 1 −ζ123 ζ123 ζ122 −ζ124 −1 −ζ124 ζ122 1 1 −ζ12 ζ125 −ζ125 ζ12 ζ124 −ζ122
1764.133.1f2 C 1 −1 ζ124 −ζ122 −ζ122 ζ124 1 ζ123 −ζ123 −ζ124 ζ122 −1 ζ122 −ζ124 1 1 ζ125 −ζ12 ζ12 −ζ125 −ζ122 ζ124
1764.133.1f3 C 1 −1 −ζ122 ζ124 ζ124 −ζ122 1 ζ123 −ζ123 ζ122 −ζ124 −1 −ζ124 ζ122 1 1 ζ12 −ζ125 ζ125 −ζ12 ζ124 −ζ122
1764.133.1f4 C 1 −1 ζ124 −ζ122 −ζ122 ζ124 1 −ζ123 ζ123 −ζ124 ζ122 −1 ζ122 −ζ124 1 1 −ζ125 ζ12 −ζ12 ζ125 −ζ122 ζ124
1764.133.2a R 2 2 −1 −1 2 2 −1 0 0 2 2 −1 −1 −1 2 2 0 0 0 0 −1 −1
1764.133.2b S 2 −2 −1 −1 2 2 −1 0 0 −2 −2 1 1 1 2 2 0 0 0 0 −1 −1
1764.133.2c1 C 2 2 −ζ3 −ζ3−1 2ζ3−1 2ζ3 −1 0 0 2ζ3 2ζ3−1 −1 −ζ3−1 −ζ3 2 2 0 0 0 0 −ζ3−1 −ζ3
1764.133.2c2 C 2 2 −ζ3−1 −ζ3 2ζ3 2ζ3−1 −1 0 0 2ζ3−1 2ζ3 −1 −ζ3 −ζ3−1 2 2 0 0 0 0 −ζ3 −ζ3−1
1764.133.2d1 C 2 −2 −ζ3 −ζ3−1 2ζ3−1 2ζ3 −1 0 0 −2ζ3 −2ζ3−1 1 ζ3−1 ζ3 2 2 0 0 0 0 −ζ3−1 −ζ3
1764.133.2d2 C 2 −2 −ζ3−1 −ζ3 2ζ3 2ζ3−1 −1 0 0 −2ζ3−1 −2ζ3 1 ζ3 ζ3−1 2 2 0 0 0 0 −ζ3 −ζ3−1
1764.133.12a R 12 0 6 6 0 0 0 0 0 0 0 0 0 0 5 −2 0 0 0 0 −1 −1
1764.133.12b1 C 12 0 6ζ3−1 6ζ3 0 0 0 0 0 0 0 0 0 0 5 −2 0 0 0 0 −ζ3 −ζ3−1
1764.133.12b2 C 12 0 6ζ3 6ζ3−1 0 0 0 0 0 0 0 0 0 0 5 −2 0 0 0 0 −ζ3−1 −ζ3
1764.133.36a R 36 0 0 0 0 0 0 0 0 0 0 0 0 0 −6 1 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