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Properties

Label 14T18
Degree $14$
Order $336$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $F_8:C_6$

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Show commands: Magma

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

Group action invariants

Degree $n$:  $14$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $18$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $F_8:C_6$
CHM label:   $[2^{4}]F_{21}(7)$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $2$
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)
magma: Generators(G);
 

Low degree resolvents

|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 TypeSizeOrderRepresentative
$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $7$ $2$ $( 4,11)( 5,12)( 7,14)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $7$ $2$ $( 3,10)( 5,12)( 6,13)( 7,14)$
$ 3, 3, 3, 3, 1, 1 $ $28$ $3$ $( 2, 3, 5)( 4, 7,13)( 6,11,14)( 9,10,12)$
$ 6, 3, 3, 1, 1 $ $28$ $6$ $( 2, 3, 5, 9,10,12)( 4,14,13)( 6,11, 7)$
$ 6, 3, 3, 1, 1 $ $28$ $6$ $( 2, 5,10, 9,12, 3)( 4, 6,14)( 7,11,13)$
$ 3, 3, 3, 3, 1, 1 $ $28$ $3$ $( 2, 5, 3)( 4,13, 7)( 6,14,11)( 9,12,10)$
$ 14 $ $24$ $14$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14)$
$ 7, 7 $ $24$ $7$ $( 1, 2, 3, 4,12, 6, 7)( 5,13,14, 8, 9,10,11)$
$ 6, 6, 2 $ $28$ $6$ $( 1, 2, 4, 8, 9,11)( 3, 6,12,10,13, 5)( 7,14)$
$ 6, 3, 3, 2 $ $28$ $6$ $( 1, 2, 4)( 3,13,12,10, 6, 5)( 7,14)( 8, 9,11)$
$ 6, 3, 3, 2 $ $28$ $6$ $( 1, 2, 6, 8, 9,13)( 3,10)( 4, 7, 5)(11,14,12)$
$ 6, 6, 2 $ $28$ $6$ $( 1, 2, 6, 8, 9,13)( 3,10)( 4,14, 5,11, 7,12)$
$ 14 $ $24$ $14$ $( 1, 4, 7,10,13, 2, 5, 8,11,14, 3, 6, 9,12)$
$ 7, 7 $ $24$ $7$ $( 1, 4,14,10,13, 2, 5)( 3, 6, 9,12, 8,11, 7)$
$ 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 8)( 2, 9)( 3,10)( 4,11)( 5,12)( 6,13)( 7,14)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $336=2^{4} \cdot 3 \cdot 7$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  336.210
magma: IdentifyGroup(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 3A-1 3A1 7A1 7A-1 7A1 7A-1
3 P 1A 2A 2B 2C 1A 1A 2B 2C 2B 2A 2A 2C 7A-1 7A1 14A-1 14A1
7 P 1A 2A 2B 2C 3A1 3A-1 6B1 6C-1 6B-1 6A1 6A-1 6C1 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 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 1 1 1 1
336.210.1c2 C 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 1 1 1 1
336.210.1d1 C 1 −1 −1 1 ζ3−1 ζ3 −ζ3 −ζ3−1 −ζ3 −ζ3−1 ζ3 ζ3−1 1 1 −1 −1
336.210.1d2 C 1 −1 −1 1 ζ3 ζ3−1 −ζ3−1 −ζ3 −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 ζ3−1 −ζ3 −ζ3−1 −ζ3 −ζ3−1 0 0 0 0
336.210.7c2 C 7 7 −1 −1 ζ3 ζ3−1 ζ3−1 ζ3 −ζ3−1 −ζ3 −ζ3−1 −ζ3 0 0 0 0
336.210.7d1 C 7 −7 1 −1 ζ3−1 ζ3 −ζ3 −ζ3−1 ζ3 ζ3−1 −ζ3 −ζ3−1 0 0 0 0
336.210.7d2 C 7 −7 1 −1 ζ3 ζ3−1 −ζ3−1 −ζ3 ζ3−1 ζ3 −ζ3−1 −ζ3 0 0 0 0

magma: CharacterTable(G);