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Properties

Label 14T7
14T7 1 3 1->3 8 1->8 9 1->9 13 1->13 2 4 2->4 2->4 2->9 12 2->12 5 3->5 10 3->10 11 3->11 3->13 6 4->6 4->8 4->10 4->11 5->3 7 5->7 5->9 5->12 6->8 6->8 6->12 6->13 7->9 14 7->14 8->2 8->10 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 $84$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $F_7 \times C_2$

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Copy content magma:G := TransitiveGroup(14, 7);
 

Group invariants

Abstract group:  $F_7 \times C_2$
Copy content magma:IdentifyGroup(G);
 
Order:  $84=2^{2} \cdot 3 \cdot 7$
Copy content magma:Order(G);
 
Cyclic:  no
Copy content magma:IsCyclic(G);
 
Abelian:  no
Copy content magma:IsAbelian(G);
 
Solvable:  yes
Copy content magma:IsSolvable(G);
 
Nilpotency class:   not nilpotent
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $14$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $7$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
CHM label:   $F_{42}(7)[x]2$
Parity:  $-1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $2$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,9,11)(2,4,8)(3,13,5)(6,12,10)$, $(1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)$, $(1,13)(2,12)(3,11)(4,10)(5,9)(6,8)$, $(1,3,5,7,9,11,13)(2,4,6,8,10,12,14)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$3$:  $C_3$
$4$:  $C_2^2$
$6$:  $C_6$ x 3
$12$:  $C_6\times C_2$
$42$:  $F_7$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 7: $F_7$

Low degree siblings

14T7, 28T15, 42T10 x 2

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

Malle's constant $a(G)$:     $1/6$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 3A1 3A-1 6A1 6A-1 6B1 6B-1 6C1 6C-1 7A 14A
Size 1 1 7 7 7 7 7 7 7 7 7 7 6 6
2 P 1A 1A 1A 1A 3A-1 3A1 3A1 3A-1 3A-1 3A1 3A1 3A-1 7A 7A
3 P 1A 2A 2B 2C 1A 1A 2B 2B 2A 2A 2C 2C 7A 14A
7 P 1A 2A 2B 2C 3A1 3A-1 6A1 6A-1 6B1 6B-1 6C1 6C-1 1A 2A
Type
84.7.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1
84.7.1b R 1 −1 −1 1 1 1 −1 −1 −1 −1 1 1 1 −1
84.7.1c R 1 −1 1 −1 1 1 1 1 −1 −1 −1 −1 1 −1
84.7.1d R 1 1 −1 −1 1 1 −1 −1 1 1 −1 −1 1 1
84.7.1e1 C 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 1 1
84.7.1e2 C 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 1 1
84.7.1f1 C 1 −1 −1 1 ζ3−1 ζ3 −ζ3 −ζ3−1 −ζ3−1 −ζ3 ζ3 ζ3−1 1 −1
84.7.1f2 C 1 −1 −1 1 ζ3 ζ3−1 −ζ3−1 −ζ3 −ζ3 −ζ3−1 ζ3−1 ζ3 1 −1
84.7.1g1 C 1 −1 1 −1 ζ3−1 ζ3 ζ3 ζ3−1 −ζ3−1 −ζ3 −ζ3 −ζ3−1 1 −1
84.7.1g2 C 1 −1 1 −1 ζ3 ζ3−1 ζ3−1 ζ3 −ζ3 −ζ3−1 −ζ3−1 −ζ3 1 −1
84.7.1h1 C 1 1 −1 −1 ζ3−1 ζ3 −ζ3 −ζ3−1 ζ3−1 ζ3 −ζ3 −ζ3−1 1 1
84.7.1h2 C 1 1 −1 −1 ζ3 ζ3−1 −ζ3−1 −ζ3 ζ3 ζ3−1 −ζ3−1 −ζ3 1 1
84.7.6a R 6 6 0 0 0 0 0 0 0 0 0 0 −1 −1
84.7.6b R 6 −6 0 0 0 0 0 0 0 0 0 0 −1 1

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed