Properties

Label 14T12
Order \(196\)
n \(14\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_7:D_7.C_2$

Related objects

Learn more about

Group action invariants

Degree $n$ :  $14$
Transitive number $t$ :  $12$
Group :  $C_7:D_7.C_2$
CHM label :  $1/2[D(7)^{2}]2$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (2,4,6,8,10,12,14), (1,6,13,8)(2,9,12,5)(3,4,11,10)(7,14), (1,13)(2,12)(3,11)(4,10)(5,9)(6,8)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
4:  $C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 7: None

Low degree siblings

14T12 x 3, 28T35 x 4

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, 1, 1 $ $49$ $2$ $( 3,13)( 4,14)( 5,11)( 6,12)( 7, 9)( 8,10)$
$ 7, 1, 1, 1, 1, 1, 1, 1 $ $4$ $7$ $( 2, 4, 6, 8,10,12,14)$
$ 7, 1, 1, 1, 1, 1, 1, 1 $ $4$ $7$ $( 2, 6,10,14, 4, 8,12)$
$ 7, 1, 1, 1, 1, 1, 1, 1 $ $4$ $7$ $( 2, 8,14, 6,12, 4,10)$
$ 4, 4, 4, 2 $ $49$ $4$ $( 1, 2)( 3, 4,13,14)( 5, 6,11,12)( 7, 8, 9,10)$
$ 4, 4, 4, 2 $ $49$ $4$ $( 1, 2)( 3,14,13, 4)( 5,12,11, 6)( 7,10, 9, 8)$
$ 7, 7 $ $4$ $7$ $( 1, 3, 5, 7, 9,11,13)( 2, 4, 6, 8,10,12,14)$
$ 7, 7 $ $4$ $7$ $( 1, 3, 5, 7, 9,11,13)( 2, 6,10,14, 4, 8,12)$
$ 7, 7 $ $4$ $7$ $( 1, 3, 5, 7, 9,11,13)( 2, 8,14, 6,12, 4,10)$
$ 7, 7 $ $4$ $7$ $( 1, 3, 5, 7, 9,11,13)( 2,10, 4,12, 6,14, 8)$
$ 7, 7 $ $4$ $7$ $( 1, 3, 5, 7, 9,11,13)( 2,12, 8, 4,14,10, 6)$
$ 7, 7 $ $4$ $7$ $( 1, 5, 9,13, 3, 7,11)( 2, 6,10,14, 4, 8,12)$
$ 7, 7 $ $4$ $7$ $( 1, 5, 9,13, 3, 7,11)( 2, 8,14, 6,12, 4,10)$
$ 7, 7 $ $4$ $7$ $( 1, 5, 9,13, 3, 7,11)( 2,10, 4,12, 6,14, 8)$
$ 7, 7 $ $4$ $7$ $( 1, 7,13, 5,11, 3, 9)( 2, 8,14, 6,12, 4,10)$

Group invariants

Order:  $196=2^{2} \cdot 7^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [196, 8]
Character table:   
      2  2  2  .  .  .  2  2  .  .  .  .  .  .  .  .  .
      7  2  .  2  2  2  .  .  2  2  2  2  2  2  2  2  2

        1a 2a 7a 7b 7c 4a 4b 7d 7e 7f 7g 7h 7i 7j 7k 7l
     2P 1a 1a 7b 7c 7a 2a 2a 7i 7k 7e 7h 7j 7l 7g 7f 7d
     3P 1a 2a 7c 7a 7b 4b 4a 7l 7f 7k 7j 7g 7d 7h 7e 7i
     5P 1a 2a 7b 7c 7a 4a 4b 7i 7k 7e 7h 7j 7l 7g 7f 7d
     7P 1a 2a 1a 1a 1a 4b 4a 1a 1a 1a 1a 1a 1a 1a 1a 1a

X.1      1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1
X.2      1  1  1  1  1 -1 -1  1  1  1  1  1  1  1  1  1
X.3      1 -1  1  1  1  J -J  1  1  1  1  1  1  1  1  1
X.4      1 -1  1  1  1 -J  J  1  1  1  1  1  1  1  1  1
X.5      4  .  A  C  B  .  .  D  G  I  I  G  E  H  H  F
X.6      4  .  B  A  C  .  .  F  I  H  H  I  D  G  G  E
X.7      4  .  C  B  A  .  .  E  H  G  G  H  F  I  I  D
X.8      4  .  D  E  F  .  .  C  I  H  H  I  B  G  G  A
X.9      4  .  E  F  D  .  .  B  G  I  I  G  A  H  H  C
X.10     4  .  F  D  E  .  .  A  H  G  G  H  C  I  I  B
X.11     4  .  G  H  I  .  .  I  C  A  E  F  G  D  B  H
X.12     4  .  H  I  G  .  .  G  B  C  F  D  H  E  A  I
X.13     4  .  I  G  H  .  .  H  A  B  D  E  I  F  C  G
X.14     4  .  G  H  I  .  .  I  F  E  A  C  G  B  D  H
X.15     4  .  H  I  G  .  .  G  D  F  C  B  H  A  E  I
X.16     4  .  I  G  H  .  .  H  E  D  B  A  I  C  F  G

A = -2*E(7)-E(7)^2-2*E(7)^3-2*E(7)^4-E(7)^5-2*E(7)^6
B = -E(7)-2*E(7)^2-2*E(7)^3-2*E(7)^4-2*E(7)^5-E(7)^6
C = -2*E(7)-2*E(7)^2-E(7)^3-E(7)^4-2*E(7)^5-2*E(7)^6
D = 2*E(7)^2+2*E(7)^5
E = 2*E(7)^3+2*E(7)^4
F = 2*E(7)+2*E(7)^6
G = E(7)^2+E(7)^3+E(7)^4+E(7)^5
H = E(7)+E(7)^3+E(7)^4+E(7)^6
I = E(7)+E(7)^2+E(7)^5+E(7)^6
J = -E(4)
  = -Sqrt(-1) = -i