Properties

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

Related objects

Learn more about

Group action invariants

Degree $n$ :  $7$
Transitive number $t$ :  $2$
Group :  $D_{7}$
CHM label :  $D(7) = 7:2$
Parity:  $-1$
Primitive:  Yes
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,6)(2,5)(3,4), (1,2,3,4,5,6,7)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Prime degree - none

Low degree siblings

14T2

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 $ $1$ $1$ $()$
$ 2, 2, 2, 1 $ $7$ $2$ $(2,7)(3,6)(4,5)$
$ 7 $ $2$ $7$ $(1,2,3,4,5,6,7)$
$ 7 $ $2$ $7$ $(1,3,5,7,2,4,6)$
$ 7 $ $2$ $7$ $(1,4,7,3,6,2,5)$

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