↑

Properties

Label 14T18
14T18 1 3 1->3 8 1->8 9 1->9 2 4 2->4 2->4 2->9 5 3->5 13 3->13 6 4->6 4->8 11 4->11 5->3 7 5->7 6->8 12 6->12 7->9 8->2 10 8->10 9->11 9->11 10->6 10->12 11->1 11->13 12->10 14 12->14 13->1 13->5 14->2
Degree $14$
Order $336$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $F_8:C_6$

Related objects

Downloads

Learn more

Show commands: Magma

Copy content magma:G := TransitiveGroup(14, 18);
 

Group invariants

Abstract group:  $F_8:C_6$
Copy content magma:IdentifyGroup(G);
 
Order:  $336=2^{4} \cdot 3 \cdot 7$
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 $t$:  $18$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
CHM label:   $[2^{4}]F_{21}(7)$
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:  $(1,9,11)(2,4,8)(3,13,5)(6,12,10)$, $(1,8)(2,9)(4,11)$, $(1,3,5,7,9,11,13)(2,4,6,8,10,12,14)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $C_6$
$21$:  $C_7:C_3$
$42$:  $(C_7:C_3) \times C_2$
$168$:  $C_2^3:(C_7: C_3)$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 7: $C_7:C_3$

Low degree siblings

16T712, 28T44, 42T67

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 $2^{4},1^{6}$ $7$ $2$ $4$ $( 1, 8)( 2, 9)( 5,12)( 7,14)$
2C $2^{3},1^{8}$ $7$ $2$ $3$ $( 2, 9)( 3,10)( 5,12)$
3A1 $3^{4},1^{2}$ $28$ $3$ $8$ $( 1, 7,12)( 3, 4, 6)( 5, 8,14)(10,11,13)$
3A-1 $3^{4},1^{2}$ $28$ $3$ $8$ $( 1,12, 7)( 3, 6, 4)( 5,14, 8)(10,13,11)$
6A1 $6,3^{2},2$ $28$ $6$ $10$ $( 1, 5, 7, 8,12,14)( 2, 9)( 3, 6, 4)(10,13,11)$
6A-1 $6,3^{2},2$ $28$ $6$ $10$ $( 1,14,12, 8, 7, 5)( 2, 9)( 3, 4, 6)(10,11,13)$
6B1 $6^{2},2$ $28$ $6$ $11$ $( 1,13, 9, 8, 6, 2)( 3,10)( 4,12, 7,11, 5,14)$
6B-1 $6^{2},2$ $28$ $6$ $11$ $( 1, 2, 6, 8, 9,13)( 3,10)( 4,14, 5,11, 7,12)$
6C1 $6,3^{2},1^{2}$ $28$ $6$ $9$ $( 2,12, 3, 9, 5,10)( 4,13, 7)( 6,14,11)$
6C-1 $6,3^{2},1^{2}$ $28$ $6$ $9$ $( 2,10, 5, 9, 3,12)( 4, 7,13)( 6,11,14)$
7A1 $7^{2}$ $24$ $7$ $12$ $( 1, 3, 5, 7, 9,11,13)( 2, 4, 6, 8,10,12,14)$
7A-1 $7^{2}$ $24$ $7$ $12$ $( 1,13,11, 9, 7, 5, 3)( 2,14,12,10, 8, 6, 4)$
14A1 $14$ $24$ $14$ $13$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14)$
14A-1 $14$ $24$ $14$ $13$ $( 1,14,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)$

Malle's constant $a(G)$:     $1/3$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 3A1 3A-1 6A1 6A-1 6B1 6B-1 6C1 6C-1 7A1 7A-1 14A1 14A-1
Size 1 1 7 7 28 28 28 28 28 28 28 28 24 24 24 24
2 P 1A 1A 1A 1A 3A-1 3A1 3A1 3A-1 3A-1 3A1 3A1 3A-1 7A1 7A-1 7A1 7A-1
3 P 1A 2A 2B 2C 1A 1A 2B 2B 2A 2A 2C 2C 7A-1 7A1 14A-1 14A1
7 P 1A 2A 2B 2C 3A1 3A-1 6A1 6A-1 6B1 6B-1 6C1 6C-1 1A 1A 2A 2A
Type
336.210.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
336.210.1b R 1 −1 1 −1 1 1 1 1 −1 −1 −1 −1 1 1 −1 −1
336.210.1c1 C 1 1 1 1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1 1 1 1 1
336.210.1c2 C 1 1 1 1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3 1 1 1 1
336.210.1d1 C 1 −1 1 −1 ζ3−1 ζ3 ζ3−1 ζ3 −ζ3 −ζ3−1 −ζ3 −ζ3−1 1 1 −1 −1
336.210.1d2 C 1 −1 1 −1 ζ3 ζ3−1 ζ3 ζ3−1 −ζ3−1 −ζ3 −ζ3−1 −ζ3 1 1 −1 −1
336.210.3a1 C 3 3 3 3 0 0 0 0 0 0 0 0 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72
336.210.3a2 C 3 3 3 3 0 0 0 0 0 0 0 0 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72
336.210.3b1 C 3 −3 3 −3 0 0 0 0 0 0 0 0 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 ζ7−3+1+ζ7+ζ72 −ζ7−3−ζ7−ζ72
336.210.3b2 C 3 −3 3 −3 0 0 0 0 0 0 0 0 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 −ζ7−3−ζ7−ζ72 ζ7−3+1+ζ7+ζ72
336.210.7a R 7 7 −1 −1 1 1 −1 −1 1 1 −1 −1 0 0 0 0
336.210.7b R 7 −7 −1 1 1 1 −1 −1 −1 −1 1 1 0 0 0 0
336.210.7c1 C 7 7 −1 −1 ζ3−1 ζ3 −ζ3−1 −ζ3 ζ3 ζ3−1 −ζ3 −ζ3−1 0 0 0 0
336.210.7c2 C 7 7 −1 −1 ζ3 ζ3−1 −ζ3 −ζ3−1 ζ3−1 ζ3 −ζ3−1 −ζ3 0 0 0 0
336.210.7d1 C 7 −7 −1 1 ζ3−1 ζ3 −ζ3−1 −ζ3 −ζ3 −ζ3−1 ζ3 ζ3−1 0 0 0 0
336.210.7d2 C 7 −7 −1 1 ζ3 ζ3−1 −ζ3 −ζ3−1 −ζ3−1 −ζ3 ζ3−1 ζ3 0 0 0 0

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed