Properties

Label 14T15
Order \(294\)
n \(14\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_7^2:S_3$

Related objects

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Group action invariants

Degree $n$ :  $14$
Transitive number $t$ :  $15$
Group :  $C_7^2:S_3$
CHM label :  $[7^{2}:3_{3}]2$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (2,4,6,8,10,12,14), (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14), (1,11,9)(2,4,8)(3,5,13)(6,12,10)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
6:  $S_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 7: None

Low degree siblings

21T17, 21T18, 42T56, 42T57, 42T62

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 $ $1$ $1$ $()$
$ 3, 3, 3, 3, 1, 1 $ $98$ $3$ $( 3, 5, 9)( 4,10, 6)( 7,13,11)( 8,12,14)$
$ 7, 1, 1, 1, 1, 1, 1, 1 $ $6$ $7$ $( 2, 4, 6, 8,10,12,14)$
$ 7, 1, 1, 1, 1, 1, 1, 1 $ $6$ $7$ $( 2, 8,14, 6,12, 4,10)$
$ 2, 2, 2, 2, 2, 2, 2 $ $21$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)$
$ 14 $ $21$ $14$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14)$
$ 14 $ $21$ $14$ $( 1, 2, 3, 6, 5,10, 7,14, 9, 4,11, 8,13,12)$
$ 14 $ $21$ $14$ $( 1, 2, 3,10, 5, 4, 7,12, 9, 6,11,14,13, 8)$
$ 14 $ $21$ $14$ $( 1, 2, 7, 8,13,14, 5, 6,11,12, 3, 4, 9,10)$
$ 14 $ $21$ $14$ $( 1, 2, 7,14,13,12, 5,10,11, 8, 3, 6, 9, 4)$
$ 14 $ $21$ $14$ $( 1, 2, 7,12,13, 8, 5, 4,11,14, 3,10, 9, 6)$
$ 7, 7 $ $3$ $7$ $( 1, 3, 5, 7, 9,11,13)( 2, 4, 6, 8,10,12,14)$
$ 7, 7 $ $3$ $7$ $( 1, 3, 5, 7, 9,11,13)( 2, 6,10,14, 4, 8,12)$
$ 7, 7 $ $6$ $7$ $( 1, 3, 5, 7, 9,11,13)( 2, 8,14, 6,12, 4,10)$
$ 7, 7 $ $3$ $7$ $( 1, 3, 5, 7, 9,11,13)( 2,10, 4,12, 6,14, 8)$
$ 7, 7 $ $6$ $7$ $( 1, 3, 5, 7, 9,11,13)( 2,12, 8, 4,14,10, 6)$
$ 7, 7 $ $6$ $7$ $( 1, 3, 5, 7, 9,11,13)( 2,14,12,10, 8, 6, 4)$
$ 7, 7 $ $3$ $7$ $( 1, 7,13, 5,11, 3, 9)( 2, 8,14, 6,12, 4,10)$
$ 7, 7 $ $3$ $7$ $( 1, 7,13, 5,11, 3, 9)( 2,12, 8, 4,14,10, 6)$
$ 7, 7 $ $3$ $7$ $( 1, 7,13, 5,11, 3, 9)( 2,14,12,10, 8, 6, 4)$

Group invariants

Order:  $294=2 \cdot 3 \cdot 7^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [294, 7]
Character table:   
      2  1  .  .  .  1   1   1   1   1   1   1  1  1  .  1  .  .  1  1  1
      3  1  1  .  .  .   .   .   .   .   .   .  .  .  .  .  .  .  .  .  .
      7  2  .  2  2  1   1   1   1   1   1   1  2  2  2  2  2  2  2  2  2

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

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

A = E(7)^3+E(7)^5+E(7)^6
  = (-1-Sqrt(-7))/2 = -1-b7
B = -3*E(7)-3*E(7)^2-2*E(7)^3-3*E(7)^4-2*E(7)^5-2*E(7)^6
  = (5-Sqrt(-7))/2 = 2-b7
C = -E(7)^3
D = -E(7)^5
E = -E(7)^6
F = 2*E(7)^4+E(7)^6
G = 2*E(7)^2+E(7)^3
H = 2*E(7)+E(7)^5
I = -2*E(7)-2*E(7)^2-2*E(7)^5-2*E(7)^6
J = -2*E(7)^2-2*E(7)^3-2*E(7)^4-2*E(7)^5
K = -2*E(7)-2*E(7)^3-2*E(7)^4-2*E(7)^6
L = 2*E(7)^3+2*E(7)^5+2*E(7)^6
  = -1-Sqrt(-7) = -1-i7
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)+E(7)^2+E(7)^5+2*E(7)^6
R = 2*E(7)^2+E(7)^3+E(7)^4+2*E(7)^5