Properties

Label 14T20
Order \(392\)
n \(14\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $D_7 \wr C_2$

Related objects

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Group action invariants

Degree $n$ :  $14$
Transitive number $t$ :  $20$
Group :  $D_7 \wr C_2$
CHM label :  $[D(7)^{2}]2=D(7)wr2$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (2,4,6,8,10,12,14), (2,12)(4,10)(6,8), (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
8:  $D_{4}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 7: None

Low degree siblings

14T20, 28T53 x 2, 28T54 x 2, 28T55 x 2, 28T57

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, 1, 1, 1, 1, 1, 1, 1, 1 $ $14$ $2$ $( 4,14)( 6,12)( 8,10)$
$ 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, 2, 2, 2, 1 $ $28$ $14$ $( 2, 4, 6, 8,10,12,14)( 3,13)( 5,11)( 7, 9)$
$ 7, 1, 1, 1, 1, 1, 1, 1 $ $4$ $7$ $( 2, 6,10,14, 4, 8,12)$
$ 7, 2, 2, 2, 1 $ $28$ $14$ $( 2, 6,10,14, 4, 8,12)( 3,13)( 5,11)( 7, 9)$
$ 7, 1, 1, 1, 1, 1, 1, 1 $ $4$ $7$ $( 2, 8,14, 6,12, 4,10)$
$ 7, 2, 2, 2, 1 $ $28$ $14$ $( 2, 8,14, 6,12, 4,10)( 3,13)( 5,11)( 7, 9)$
$ 2, 2, 2, 2, 2, 2, 2 $ $14$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)$
$ 4, 4, 4, 2 $ $98$ $4$ $( 1, 2)( 3, 4,13,14)( 5, 6,11,12)( 7, 8, 9,10)$
$ 14 $ $28$ $14$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14)$
$ 14 $ $28$ $14$ $( 1, 2, 5, 6, 9,10,13,14, 3, 4, 7, 8,11,12)$
$ 14 $ $28$ $14$ $( 1, 2, 7, 8,13,14, 5, 6,11,12, 3, 4, 9,10)$
$ 7, 7 $ $4$ $7$ $( 1, 3, 5, 7, 9,11,13)( 2, 4, 6, 8,10,12,14)$
$ 7, 7 $ $8$ $7$ $( 1, 3, 5, 7, 9,11,13)( 2, 6,10,14, 4, 8,12)$
$ 7, 7 $ $8$ $7$ $( 1, 3, 5, 7, 9,11,13)( 2, 8,14, 6,12, 4,10)$
$ 7, 7 $ $4$ $7$ $( 1, 5, 9,13, 3, 7,11)( 2, 6,10,14, 4, 8,12)$
$ 7, 7 $ $8$ $7$ $( 1, 5, 9,13, 3, 7,11)( 2, 8,14, 6,12, 4,10)$
$ 7, 7 $ $4$ $7$ $( 1, 7,13, 5,11, 3, 9)( 2, 8,14, 6,12, 4,10)$

Group invariants

Order:  $392=2^{3} \cdot 7^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [392, 37]
Character table:   
      2  3  2  3  1   1  1   1  1   1  2  2   1   1   1  1  .  .  1  .  1
      7  2  1  .  2   1  2   1  2   1  1  .   1   1   1  2  2  2  2  2  2

        1a 2a 2b 7a 14a 7b 14b 7c 14c 2c 4a 14d 14e 14f 7d 7e 7f 7g 7h 7i
     2P 1a 1a 1a 7b  7b 7c  7c 7a  7a 1a 2b  7d  7g  7i 7g 7h 7e 7i 7f 7d
     3P 1a 2a 2b 7c 14c 7a 14a 7b 14b 2c 4a 14f 14d 14e 7i 7f 7h 7d 7e 7g
     5P 1a 2a 2b 7b 14b 7c 14c 7a 14a 2c 4a 14e 14f 14d 7g 7h 7e 7i 7f 7d
     7P 1a 2a 2b 1a  2a 1a  2a 1a  2a 2c 4a  2c  2c  2c 1a 1a 1a 1a 1a 1a
    11P 1a 2a 2b 7c 14c 7a 14a 7b 14b 2c 4a 14f 14d 14e 7i 7f 7h 7d 7e 7g
    13P 1a 2a 2b 7a 14a 7b 14b 7c 14c 2c 4a 14d 14e 14f 7d 7e 7f 7g 7h 7i

X.1      1  1  1  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  1  1  1  1
X.3      1 -1  1  1  -1  1  -1  1  -1  1 -1   1   1   1  1  1  1  1  1  1
X.4      1  1  1  1   1  1   1  1   1 -1 -1  -1  -1  -1  1  1  1  1  1  1
X.5      2  . -2  2   .  2   .  2   .  .  .   .   .   .  2  2  2  2  2  2
X.6      4 -2  .  A   J  C   L  B   K  .  .   .   .   .  D  M  N  F  O  E
X.7      4 -2  .  B   K  A   J  C   L  .  .   .   .   .  E  N  O  D  M  F
X.8      4 -2  .  C   L  B   K  A   J  .  .   .   .   .  F  O  M  E  N  D
X.9      4  .  .  D   .  F   .  E   . -2  .   J   L   K  C  N  O  B  M  A
X.10     4  .  .  E   .  D   .  F   . -2  .   K   J   L  A  O  M  C  N  B
X.11     4  .  .  F   .  E   .  D   . -2  .   L   K   J  B  M  N  A  O  C
X.12     4  .  .  D   .  F   .  E   .  2  .  -J  -L  -K  C  N  O  B  M  A
X.13     4  .  .  E   .  D   .  F   .  2  .  -K  -J  -L  A  O  M  C  N  B
X.14     4  .  .  F   .  E   .  D   .  2  .  -L  -K  -J  B  M  N  A  O  C
X.15     4  2  .  A  -J  C  -L  B  -K  .  .   .   .   .  D  M  N  F  O  E
X.16     4  2  .  B  -K  A  -J  C  -L  .  .   .   .   .  E  N  O  D  M  F
X.17     4  2  .  C  -L  B  -K  A  -J  .  .   .   .   .  F  O  M  E  N  D
X.18     8  .  .  G   .  I   .  H   .  .  .   .   .   .  H  P  Q  G  R  I
X.19     8  .  .  H   .  G   .  I   .  .  .   .   .   .  I  Q  R  H  P  G
X.20     8  .  .  I   .  H   .  G   .  .  .   .   .   .  G  R  P  I  Q  H

A = -2*E(7)-2*E(7)^2-E(7)^3-E(7)^4-2*E(7)^5-2*E(7)^6
B = -2*E(7)-E(7)^2-2*E(7)^3-2*E(7)^4-E(7)^5-2*E(7)^6
C = -E(7)-2*E(7)^2-2*E(7)^3-2*E(7)^4-2*E(7)^5-E(7)^6
D = 2*E(7)^3+2*E(7)^4
E = 2*E(7)^2+2*E(7)^5
F = 2*E(7)+2*E(7)^6
G = 2*E(7)+2*E(7)^3+2*E(7)^4+2*E(7)^6
H = 2*E(7)^2+2*E(7)^3+2*E(7)^4+2*E(7)^5
I = 2*E(7)+2*E(7)^2+2*E(7)^5+2*E(7)^6
J = -E(7)^3-E(7)^4
K = -E(7)^2-E(7)^5
L = -E(7)-E(7)^6
M = E(7)+E(7)^3+E(7)^4+E(7)^6
N = E(7)^2+E(7)^3+E(7)^4+E(7)^5
O = E(7)+E(7)^2+E(7)^5+E(7)^6
P = -E(7)-2*E(7)^3-2*E(7)^4-E(7)^6
Q = -2*E(7)^2-E(7)^3-E(7)^4-2*E(7)^5
R = -2*E(7)-E(7)^2-E(7)^5-2*E(7)^6