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Properties

Label 14T8
14T8 1 8 1->8 2 4 2->4 9 2->9 3 10 3->10 6 4->6 11 4->11 5 12 5->12 6->8 13 6->13 7 14 7->14 8->10 10->12 12->14 14->2
Degree $14$
Order $98$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_7 \wr C_2$

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Copy content magma:G := TransitiveGroup(14, 8);
 

Group invariants

Abstract group:  $C_7 \wr C_2$
Copy content magma:IdentifyGroup(G);
 
Order:  $98=2 \cdot 7^{2}$
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 $t$:  $8$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
CHM label:   $[7^{2}]2=7wr2$
Parity:  $-1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $7$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(2,4,6,8,10,12,14)$, $(1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$7$:  $C_7$
$14$:  $D_{7}$, $C_{14}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 7: None

Low degree siblings

14T8 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 $2^{7}$ $7$ $2$ $7$ $( 1, 8)( 2, 9)( 3,10)( 4,11)( 5,12)( 6,13)( 7,14)$
7A1 $7^{2}$ $1$ $7$ $12$ $( 1, 9, 3,11, 5,13, 7)( 2,10, 4,12, 6,14, 8)$
7A-1 $7^{2}$ $1$ $7$ $12$ $( 1, 7,13, 5,11, 3, 9)( 2, 8,14, 6,12, 4,10)$
7A2 $7^{2}$ $1$ $7$ $12$ $( 1, 3, 5, 7, 9,11,13)( 2, 4, 6, 8,10,12,14)$
7A-2 $7^{2}$ $1$ $7$ $12$ $( 1,13,11, 9, 7, 5, 3)( 2,14,12,10, 8, 6, 4)$
7A3 $7^{2}$ $1$ $7$ $12$ $( 1,11, 7, 3,13, 9, 5)( 2,12, 8, 4,14,10, 6)$
7A-3 $7^{2}$ $1$ $7$ $12$ $( 1, 5, 9,13, 3, 7,11)( 2, 6,10,14, 4, 8,12)$
7B1 $7^{2}$ $2$ $7$ $12$ $( 1, 3, 5, 7, 9,11,13)( 2,14,12,10, 8, 6, 4)$
7B2 $7^{2}$ $2$ $7$ $12$ $( 1, 5, 9,13, 3, 7,11)( 2,12, 8, 4,14,10, 6)$
7B3 $7^{2}$ $2$ $7$ $12$ $( 1, 7,13, 5,11, 3, 9)( 2,10, 4,12, 6,14, 8)$
7C1 $7^{2}$ $2$ $7$ $12$ $( 1,11, 7, 3,13, 9, 5)( 2, 8,14, 6,12, 4,10)$
7C-1 $7^{2}$ $2$ $7$ $12$ $( 1, 9, 3,11, 5,13, 7)( 2, 6,10,14, 4, 8,12)$
7C2 $7^{2}$ $2$ $7$ $12$ $( 1, 7,13, 5,11, 3, 9)( 2,14,12,10, 8, 6, 4)$
7C-2 $7^{2}$ $2$ $7$ $12$ $( 1, 3, 5, 7, 9,11,13)( 2,10, 4,12, 6,14, 8)$
7C3 $7^{2}$ $2$ $7$ $12$ $( 1, 3, 5, 7, 9,11,13)( 2, 6,10,14, 4, 8,12)$
7C-3 $7^{2}$ $2$ $7$ $12$ $( 1,11, 7, 3,13, 9, 5)( 2,14,12,10, 8, 6, 4)$
7D1 $7^{2}$ $2$ $7$ $12$ $( 1,13,11, 9, 7, 5, 3)( 2, 6,10,14, 4, 8,12)$
7D-1 $7^{2}$ $2$ $7$ $12$ $( 1,11, 7, 3,13, 9, 5)( 2, 4, 6, 8,10,12,14)$
7D2 $7^{2}$ $2$ $7$ $12$ $( 1, 9, 3,11, 5,13, 7)( 2,12, 8, 4,14,10, 6)$
7D-2 $7^{2}$ $2$ $7$ $12$ $( 1, 5, 9,13, 3, 7,11)( 2, 8,14, 6,12, 4,10)$
7D3 $7^{2}$ $2$ $7$ $12$ $( 1,13,11, 9, 7, 5, 3)( 2,10, 4,12, 6,14, 8)$
7D-3 $7^{2}$ $2$ $7$ $12$ $( 1, 7,13, 5,11, 3, 9)( 2, 4, 6, 8,10,12,14)$
7E1 $7,1^{7}$ $2$ $7$ $6$ $( 1, 5, 9,13, 3, 7,11)$
7E-1 $7,1^{7}$ $2$ $7$ $6$ $( 2,12, 8, 4,14,10, 6)$
7E2 $7,1^{7}$ $2$ $7$ $6$ $( 1, 9, 3,11, 5,13, 7)$
7E-2 $7,1^{7}$ $2$ $7$ $6$ $( 2, 8,14, 6,12, 4,10)$
7E3 $7,1^{7}$ $2$ $7$ $6$ $( 1,13,11, 9, 7, 5, 3)$
7E-3 $7,1^{7}$ $2$ $7$ $6$ $( 2, 4, 6, 8,10,12,14)$
14A1 $14$ $7$ $14$ $13$ $( 1,12, 9, 6, 3,14,11, 8, 5, 2,13,10, 7, 4)$
14A-1 $14$ $7$ $14$ $13$ $( 1, 4, 7,10,13, 2, 5, 8,11,14, 3, 6, 9,12)$
14A3 $14$ $7$ $14$ $13$ $( 1, 6,11, 2, 7,12, 3, 8,13, 4, 9,14, 5,10)$
14A-3 $14$ $7$ $14$ $13$ $( 1,10, 5,14, 9, 4,13, 8, 3,12, 7, 2,11, 6)$
14A5 $14$ $7$ $14$ $13$ $( 1,14,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)$
14A-5 $14$ $7$ $14$ $13$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14)$

Malle's constant $a(G)$:     $1/6$

Copy content magma:ConjugacyClasses(G);
 

Character table

35 x 35 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed