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Properties

Label 14T23
14T23 1 6 1->6 9 1->9 13 1->13 2 4 2->4 2->4 2->9 12 2->12 3 3->4 11 3->11 3->13 4->6 8 4->8 10 4->10 4->11 5 5->2 5->3 5->9 6->8 6->8 6->12 6->13 7 14 7->14 8->1 8->2 8->10 9->11 9->12 10->3 10->6 10->12 11->1 11->10 12->5 12->10 12->14 13->5 13->8 14->2
Degree $14$
Order $588$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_7^2:C_{12}$

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Show commands: Magma

Copy content magma:G := TransitiveGroup(14, 23);
 

Group invariants

Abstract group:  $C_7^2:C_{12}$
Copy content magma:IdentifyGroup(G);
 
Order:  $588=2^{2} \cdot 3 \cdot 7^{2}$
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 $t$:  $23$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
CHM label:   $[1/6_+.F_{42}(7)^{2}]2_{2}$
Parity:  $1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $1$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(2,4,6,8,10,12,14)$, $(1,9,11)(2,4,8)(3,13,5)(6,12,10)$, $(1,6,13,8)(2,9,12,5)(3,4,11,10)(7,14)$, $(1,13)(2,12)(3,11)(4,10)(5,9)(6,8)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$4$:  $C_4$
$6$:  $C_6$
$12$:  $C_{12}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 7: None

Low degree siblings

14T23 x 3, 28T76 x 4, 42T120 x 4

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 $2^{6},1^{2}$ $49$ $2$ $6$ $( 1,13)( 2,10)( 3,11)( 4, 8)( 5, 9)(12,14)$
3A1 $3^{4},1^{2}$ $49$ $3$ $8$ $( 1,11, 9)( 2, 4,12)( 3, 5,13)( 8,14,10)$
3A-1 $3^{4},1^{2}$ $49$ $3$ $8$ $( 1, 9,11)( 2,12, 4)( 3,13, 5)( 8,10,14)$
4A1 $4^{3},2$ $49$ $4$ $10$ $( 1,12,13,14)( 2, 3,10,11)( 4, 5, 8, 9)( 6, 7)$
4A-1 $4^{3},2$ $49$ $4$ $10$ $( 1,14,13,12)( 2,11,10, 3)( 4, 9, 8, 5)( 6, 7)$
6A1 $6^{2},1^{2}$ $49$ $6$ $10$ $( 1, 5,11,13, 9, 3)( 2,14, 4,10,12, 8)$
6A-1 $6^{2},1^{2}$ $49$ $6$ $10$ $( 1, 3, 9,13,11, 5)( 2, 8,12,10, 4,14)$
7A $7^{2}$ $12$ $7$ $12$ $( 1, 5, 9,13, 3, 7,11)( 2, 8,14, 6,12, 4,10)$
7B $7,1^{7}$ $12$ $7$ $6$ $( 2, 4, 6, 8,10,12,14)$
7C $7^{2}$ $12$ $7$ $12$ $( 1, 5, 9,13, 3, 7,11)( 2,10, 4,12, 6,14, 8)$
7D $7^{2}$ $12$ $7$ $12$ $( 1, 5, 9,13, 3, 7,11)( 2,12, 8, 4,14,10, 6)$
12A1 $12,2$ $49$ $12$ $12$ $( 1,10, 5,12,11, 8,13, 2, 9,14, 3, 4)( 6, 7)$
12A-1 $12,2$ $49$ $12$ $12$ $( 1, 4, 3,14, 9, 2,13, 8,11,12, 5,10)( 6, 7)$
12A5 $12,2$ $49$ $12$ $12$ $( 1, 8, 3,12, 9,10,13, 4,11,14, 5, 2)( 6, 7)$
12A-5 $12,2$ $49$ $12$ $12$ $( 1, 2, 5,14,11, 4,13,10, 9,12, 3, 8)( 6, 7)$

Malle's constant $a(G)$:     $1/6$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 3A1 3A-1 4A1 4A-1 6A1 6A-1 7A 7B 7C 7D 12A1 12A-1 12A5 12A-5
Size 1 49 49 49 49 49 49 49 12 12 12 12 49 49 49 49
2 P 1A 1A 3A-1 3A1 2A 2A 3A1 3A-1 7A 7B 7C 7D 6A1 6A-1 6A-1 6A1
3 P 1A 2A 1A 1A 4A-1 4A1 2A 2A 7A 7B 7C 7D 4A1 4A-1 4A1 4A-1
7 P 1A 2A 3A1 3A-1 4A-1 4A1 6A1 6A-1 1A 1A 1A 1A 12A-5 12A5 12A-1 12A1
Type
588.34.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
588.34.1b R 1 1 1 1 −1 −1 1 1 1 1 1 1 −1 −1 −1 −1
588.34.1c1 C 1 1 ζ3−1 ζ3 1 1 ζ3 ζ3−1 1 1 1 1 ζ3 ζ3 ζ3−1 ζ3−1
588.34.1c2 C 1 1 ζ3 ζ3−1 1 1 ζ3−1 ζ3 1 1 1 1 ζ3−1 ζ3−1 ζ3 ζ3
588.34.1d1 C 1 −1 1 1 −i i −1 −1 1 1 1 1 i −i −i i
588.34.1d2 C 1 −1 1 1 i −i −1 −1 1 1 1 1 −i i i −i
588.34.1e1 C 1 1 ζ3−1 ζ3 −1 −1 ζ3 ζ3−1 1 1 1 1 −ζ3 −ζ3 −ζ3−1 −ζ3−1
588.34.1e2 C 1 1 ζ3 ζ3−1 −1 −1 ζ3−1 ζ3 1 1 1 1 −ζ3−1 −ζ3−1 −ζ3 −ζ3
588.34.1f1 C 1 −1 −ζ122 ζ124 −ζ123 ζ123 −ζ124 ζ122 1 1 1 1 −ζ12 ζ12 ζ125 −ζ125
588.34.1f2 C 1 −1 ζ124 −ζ122 ζ123 −ζ123 ζ122 −ζ124 1 1 1 1 ζ125 −ζ125 −ζ12 ζ12
588.34.1f3 C 1 −1 −ζ122 ζ124 ζ123 −ζ123 −ζ124 ζ122 1 1 1 1 ζ12 −ζ12 −ζ125 ζ125
588.34.1f4 C 1 −1 ζ124 −ζ122 −ζ123 ζ123 ζ122 −ζ124 1 1 1 1 −ζ125 ζ125 ζ12 −ζ12
588.34.12a R 12 0 0 0 0 0 0 0 −2 −2 −2 5 0 0 0 0
588.34.12b R 12 0 0 0 0 0 0 0 −2 −2 5 −2 0 0 0 0
588.34.12c R 12 0 0 0 0 0 0 0 −2 5 −2 −2 0 0 0 0
588.34.12d R 12 0 0 0 0 0 0 0 5 −2 −2 −2 0 0 0 0

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed