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Properties

Label 14T1
Degree $14$
Order $14$
Cyclic yes
Abelian yes
Solvable yes
Primitive no
$p$-group no
Group: $C_{14}$

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Show commands: Magma

magma: G := TransitiveGroup(14, 1);
 

Group action invariants

Degree $n$:  $14$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $1$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_{14}$
CHM label:   $C(14)=7[x]2$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $14$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,2,3,4,5,6,7,8,9,10,11,12,13,14)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$7$:  $C_7$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 7: $C_7$

Low degree siblings

There are no siblings 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 $ $1$ $1$ $()$
$ 14 $ $1$ $14$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14)$
$ 7, 7 $ $1$ $7$ $( 1, 3, 5, 7, 9,11,13)( 2, 4, 6, 8,10,12,14)$
$ 14 $ $1$ $14$ $( 1, 4, 7,10,13, 2, 5, 8,11,14, 3, 6, 9,12)$
$ 7, 7 $ $1$ $7$ $( 1, 5, 9,13, 3, 7,11)( 2, 6,10,14, 4, 8,12)$
$ 14 $ $1$ $14$ $( 1, 6,11, 2, 7,12, 3, 8,13, 4, 9,14, 5,10)$
$ 7, 7 $ $1$ $7$ $( 1, 7,13, 5,11, 3, 9)( 2, 8,14, 6,12, 4,10)$
$ 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 8)( 2, 9)( 3,10)( 4,11)( 5,12)( 6,13)( 7,14)$
$ 7, 7 $ $1$ $7$ $( 1, 9, 3,11, 5,13, 7)( 2,10, 4,12, 6,14, 8)$
$ 14 $ $1$ $14$ $( 1,10, 5,14, 9, 4,13, 8, 3,12, 7, 2,11, 6)$
$ 7, 7 $ $1$ $7$ $( 1,11, 7, 3,13, 9, 5)( 2,12, 8, 4,14,10, 6)$
$ 14 $ $1$ $14$ $( 1,12, 9, 6, 3,14,11, 8, 5, 2,13,10, 7, 4)$
$ 7, 7 $ $1$ $7$ $( 1,13,11, 9, 7, 5, 3)( 2,14,12,10, 8, 6, 4)$
$ 14 $ $1$ $14$ $( 1,14,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $14=2 \cdot 7$
magma: Order(G);
 
Cyclic:  yes
magma: IsCyclic(G);
 
Abelian:  yes
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:  $1$
Label:  14.2
magma: IdentifyGroup(G);
 
Character table:

1A 2A 7A1 7A-1 7A2 7A-2 7A3 7A-3 14A1 14A-1 14A3 14A-3 14A5 14A-5
Size 1 1 1 1 1 1 1 1 1 1 1 1 1 1
2 P 1A 1A 7A2 7A-2 7A-3 7A3 7A-1 7A1 7A-3 7A-1 7A1 7A3 7A-2 7A2
7 P 1A 2A 7A3 7A-3 7A-1 7A1 7A2 7A-2 14A5 14A-3 14A3 14A-5 14A1 14A-1
Type
14.2.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1
14.2.1b R 1 −1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1
14.2.1c1 C 1 1 ζ7−3 ζ73 ζ7 ζ7−1 ζ7−2 ζ72 ζ72 ζ7−2 ζ7−1 ζ7 ζ73 ζ7−3
14.2.1c2 C 1 1 ζ73 ζ7−3 ζ7−1 ζ7 ζ72 ζ7−2 ζ7−2 ζ72 ζ7 ζ7−1 ζ7−3 ζ73
14.2.1c3 C 1 1 ζ7−2 ζ72 ζ73 ζ7−3 ζ7 ζ7−1 ζ7−1 ζ7 ζ7−3 ζ73 ζ72 ζ7−2
14.2.1c4 C 1 1 ζ72 ζ7−2 ζ7−3 ζ73 ζ7−1 ζ7 ζ7 ζ7−1 ζ73 ζ7−3 ζ7−2 ζ72
14.2.1c5 C 1 1 ζ7−1 ζ7 ζ7−2 ζ72 ζ7−3 ζ73 ζ73 ζ7−3 ζ72 ζ7−2 ζ7 ζ7−1
14.2.1c6 C 1 1 ζ7 ζ7−1 ζ72 ζ7−2 ζ73 ζ7−3 ζ7−3 ζ73 ζ7−2 ζ72 ζ7−1 ζ7
14.2.1d1 C 1 −1 ζ7−3 ζ73 ζ7 ζ7−1 ζ7−2 ζ72 −ζ72 −ζ7−2 −ζ7−1 −ζ7 −ζ73 −ζ7−3
14.2.1d2 C 1 −1 ζ73 ζ7−3 ζ7−1 ζ7 ζ72 ζ7−2 −ζ7−2 −ζ72 −ζ7 −ζ7−1 −ζ7−3 −ζ73
14.2.1d3 C 1 −1 ζ7−2 ζ72 ζ73 ζ7−3 ζ7 ζ7−1 −ζ7−1 −ζ7 −ζ7−3 −ζ73 −ζ72 −ζ7−2
14.2.1d4 C 1 −1 ζ72 ζ7−2 ζ7−3 ζ73 ζ7−1 ζ7 −ζ7 −ζ7−1 −ζ73 −ζ7−3 −ζ7−2 −ζ72
14.2.1d5 C 1 −1 ζ7−1 ζ7 ζ7−2 ζ72 ζ7−3 ζ73 −ζ73 −ζ7−3 −ζ72 −ζ7−2 −ζ7 −ζ7−1
14.2.1d6 C 1 −1 ζ7 ζ7−1 ζ72 ζ7−2 ζ73 ζ7−3 −ζ7−3 −ζ73 −ζ7−2 −ζ72 −ζ7−1 −ζ7

magma: CharacterTable(G);