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Properties

Label 14T24
14T24 1 8 1->8 9 1->9 13 1->13 2 4 2->4 2->4 2->9 12 2->12 3 10 3->10 11 3->11 3->13 6 4->6 4->8 4->10 4->11 5 5->3 5->9 5->12 6->8 6->8 6->12 6->13 7 14 7->14 8->2 8->10 9->11 10->6 10->12 11->1 12->10 12->14 13->5 14->2
Degree $14$
Order $588$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_7:F_7$

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Copy content magma:G := TransitiveGroup(14, 24);
 

Group invariants

Abstract group:  $D_7:F_7$
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 $t$:  $24$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
CHM label:   $[7^{2}:6]2$
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:  $(2,4,6,8,10,12,14)$, $(1,9,11)(2,4,8)(3,13,5)(6,12,10)$, $(1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)$, $(1,13)(2,12)(3,11)(4,10)(5,9)(6,8)$
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
$12$:  $C_6\times C_2$
$42$:  $F_7$ x 2
$84$:  $F_7 \times C_2$ x 2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 7: None

Low degree siblings

14T24 x 2, 28T77 x 3, 42T121 x 3

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 $2^{7}$ $7$ $2$ $7$ $( 1, 6)( 2,11)( 3, 8)( 4,13)( 5,10)( 7,12)( 9,14)$
2B $2^{7}$ $7$ $2$ $7$ $( 1,14)( 2,13)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)$
2C $2^{6},1^{2}$ $49$ $2$ $6$ $( 1, 7)( 2,14)( 3, 5)( 4,12)( 6,10)( 9,13)$
3A1 $3^{4},1^{2}$ $49$ $3$ $8$ $( 1, 5,13)( 3, 9, 7)( 4, 6,10)( 8,14,12)$
3A-1 $3^{4},1^{2}$ $49$ $3$ $8$ $( 1,13, 5)( 3, 7, 9)( 4,10, 6)( 8,12,14)$
6A1 $6^{2},2$ $49$ $6$ $11$ $( 1, 4, 5, 6,13,10)( 2,11)( 3,12, 9, 8, 7,14)$
6A-1 $6^{2},2$ $49$ $6$ $11$ $( 1,10,13, 6, 5, 4)( 2,11)( 3,14, 7, 8, 9,12)$
6B1 $6^{2},2$ $49$ $6$ $11$ $( 1,12, 7,14, 3, 8)( 2,13)( 4, 9,10,11, 6, 5)$
6B-1 $6^{2},2$ $49$ $6$ $11$ $( 1, 8, 3,14, 7,12)( 2,13)( 4, 5, 6,11,10, 9)$
6C1 $6^{2},1^{2}$ $49$ $6$ $10$ $( 1, 9, 5, 7,13, 3)( 2, 4,10,14,12, 6)$
6C-1 $6^{2},1^{2}$ $49$ $6$ $10$ $( 1, 3,13, 7, 5, 9)( 2, 6,12,14,10, 4)$
7A $7^{2}$ $6$ $7$ $12$ $( 1,11, 7, 3,13, 9, 5)( 2, 6,10,14, 4, 8,12)$
7B $7^{2}$ $6$ $7$ $12$ $( 1, 7,13, 5,11, 3, 9)( 2, 8,14, 6,12, 4,10)$
7C $7,1^{7}$ $12$ $7$ $6$ $( 2,10, 4,12, 6,14, 8)$
7D $7^{2}$ $12$ $7$ $12$ $( 1,11, 7, 3,13, 9, 5)( 2,14,12,10, 8, 6, 4)$
7E $7^{2}$ $12$ $7$ $12$ $( 1, 5, 9,13, 3, 7,11)( 2,14,12,10, 8, 6, 4)$
14A $14$ $42$ $14$ $13$ $( 1, 6,11,10, 7,14, 3, 4,13, 8, 9,12, 5, 2)$
14B $14$ $42$ $14$ $13$ $( 1, 4, 7,10,13, 2, 5, 8,11,14, 3, 6, 9,12)$

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 7B 7C 7D 7E 14A 14B
Size 1 7 7 49 49 49 49 49 49 49 49 49 6 6 12 12 12 42 42
2 P 1A 1A 1A 1A 3A-1 3A1 3A1 3A-1 3A-1 3A1 3A1 3A-1 7A 7B 7C 7D 7E 7A 7B
3 P 1A 2A 2B 2C 1A 1A 2A 2A 2B 2B 2C 2C 7A 7B 7C 7D 7E 14A 14B
7 P 1A 2A 2B 2C 3A1 3A-1 6A1 6A-1 6B1 6B-1 6C1 6C-1 1A 1A 1A 1A 1A 2B 2A
Type
588.37.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
588.37.1b R 1 −1 −1 1 1 1 −1 −1 1 1 −1 −1 1 1 1 1 1 −1 −1
588.37.1c R 1 −1 1 −1 1 1 1 1 −1 −1 −1 −1 1 1 1 1 1 1 −1
588.37.1d R 1 1 −1 −1 1 1 −1 −1 −1 −1 1 1 1 1 1 1 1 −1 1
588.37.1e1 C 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3 1 1 1 1 1 1 1
588.37.1e2 C 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3 ζ3−1 1 1 1 1 1 1 1
588.37.1f1 C 1 −1 −1 1 ζ3−1 ζ3 −ζ3 −ζ3−1 ζ3 ζ3−1 −ζ3−1 −ζ3 1 1 1 1 1 −1 −1
588.37.1f2 C 1 −1 −1 1 ζ3 ζ3−1 −ζ3−1 −ζ3 ζ3−1 ζ3 −ζ3 −ζ3−1 1 1 1 1 1 −1 −1
588.37.1g1 C 1 −1 1 −1 ζ3−1 ζ3 ζ3 ζ3−1 −ζ3 −ζ3−1 −ζ3−1 −ζ3 1 1 1 1 1 1 −1
588.37.1g2 C 1 −1 1 −1 ζ3 ζ3−1 ζ3−1 ζ3 −ζ3−1 −ζ3 −ζ3 −ζ3−1 1 1 1 1 1 1 −1
588.37.1h1 C 1 1 −1 −1 ζ3−1 ζ3 −ζ3 −ζ3−1 −ζ3 −ζ3−1 ζ3−1 ζ3 1 1 1 1 1 −1 1
588.37.1h2 C 1 1 −1 −1 ζ3 ζ3−1 −ζ3−1 −ζ3 −ζ3−1 −ζ3 ζ3 ζ3−1 1 1 1 1 1 −1 1
588.37.6a R 6 0 6 0 0 0 0 0 0 0 0 0 −1 6 −1 −1 −1 −1 0
588.37.6b R 6 6 0 0 0 0 0 0 0 0 0 0 6 −1 −1 −1 −1 0 −1
588.37.6c R 6 −6 0 0 0 0 0 0 0 0 0 0 6 −1 −1 −1 −1 0 1
588.37.6d R 6 0 −6 0 0 0 0 0 0 0 0 0 −1 6 −1 −1 −1 1 0
588.37.12a R 12 0 0 0 0 0 0 0 0 0 0 0 −2 −2 −2 −2 5 0 0
588.37.12b R 12 0 0 0 0 0 0 0 0 0 0 0 −2 −2 −2 5 −2 0 0
588.37.12c R 12 0 0 0 0 0 0 0 0 0 0 0 −2 −2 5 −2 −2 0 0

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed