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Properties

Label 14T2
Order \(14\)
n \(14\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $D_{7}$

Related objects

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Group action invariants

Degree $n$ :  $14$
Transitive number $t$ :  $2$
Group :  $D_{7}$
CHM label :  $D_{14}(14)=[7]2$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,14), (1,3,5,7,9,11,13)(2,4,6,8,10,12,14)
$|\Aut(F/K)|$:  $14$

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 7: $D_{7}$

Low degree siblings

7T2

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy Classes

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

Group invariants

Order:  $14=2 \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [14, 1]
Character table:    
     2  1  1  .  .  .
     7  1  .  1  1  1

       1a 2a 7a 7b 7c
    2P 1a 1a 7b 7c 7a
    3P 1a 2a 7c 7a 7b
    5P 1a 2a 7b 7c 7a
    7P 1a 2a 1a 1a 1a

X.1     1  1  1  1  1
X.2     1 -1  1  1  1
X.3     2  .  A  B  C
X.4     2  .  B  C  A
X.5     2  .  C  A  B

A = E(7)+E(7)^6
B = E(7)^2+E(7)^5
C = E(7)^3+E(7)^4