Properties

Label 14T14
Order \(294\)
n \(14\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_7^2:C_6$

Related objects

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Group action invariants

Degree $n$ :  $14$
Transitive number $t$ :  $14$
Group :  $C_7^2:C_6$
CHM label :  $[7^{2}:3]2$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (2,4,6,8,10,12,14), (1,9,11)(2,4,8)(3,13,5)(6,12,10), (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:  $C_2$
3:  $C_3$
6:  $C_6$
21:  $C_7:C_3$
42:  $F_7$, $(C_7:C_3) \times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 7: None

Low degree siblings

14T14 x 2, 42T61 x 3

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 $ $49$ $3$ $( 3, 5, 9)( 4, 6,10)( 7,13,11)( 8,14,12)$
$ 3, 3, 3, 3, 1, 1 $ $49$ $3$ $( 3, 9, 5)( 4,10, 6)( 7,11,13)( 8,12,14)$
$ 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 $ $7$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)$
$ 6, 6, 2 $ $49$ $6$ $( 1, 2)( 3, 6, 9, 4, 5,10)( 7,14,11, 8,13,12)$
$ 6, 6, 2 $ $49$ $6$ $( 1, 2)( 3,10, 5, 4, 9, 6)( 7,12,13, 8,11,14)$
$ 14 $ $21$ $14$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14)$
$ 14 $ $21$ $14$ $( 1, 2, 7, 8,13,14, 5, 6,11,12, 3, 4, 9,10)$
$ 7, 7 $ $3$ $7$ $( 1, 3, 5, 7, 9,11,13)( 2, 4, 6, 8,10,12,14)$
$ 7, 7 $ $6$ $7$ $( 1, 3, 5, 7, 9,11,13)( 2, 6,10,14, 4, 8,12)$
$ 7, 7 $ $6$ $7$ $( 1, 3, 5, 7, 9,11,13)( 2, 8,14, 6,12, 4,10)$
$ 7, 7 $ $6$ $7$ $( 1, 3, 5, 7, 9,11,13)( 2,12, 8, 4,14,10, 6)$
$ 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 $ $6$ $7$ $( 1, 7,13, 5,11, 3, 9)( 2,12, 8, 4,14,10, 6)$

Group invariants

Order:  $294=2 \cdot 3 \cdot 7^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [294, 12]
Character table:   
      2  1  1  1  .  .  1   1   1   1   1  1  .  .  .  .  1  .
      3  1  1  1  .  .  1   1   1   .   .  .  .  .  .  .  .  .
      7  2  .  .  2  2  1   .   .   1   1  2  2  2  2  2  2  2

        1a 3a 3b 7a 7b 2a  6a  6b 14a 14b 7c 7d 7e 7f 7g 7h 7i
     2P 1a 3b 3a 7a 7b 1a  3b  3a  7c  7h 7c 7d 7e 7f 7g 7h 7i
     3P 1a 1a 1a 7b 7a 2a  2a  2a 14b 14a 7h 7i 7f 7e 7g 7c 7d
     5P 1a 3b 3a 7b 7a 2a  6b  6a 14b 14a 7h 7i 7f 7e 7g 7c 7d
     7P 1a 3a 3b 1a 1a 2a  6a  6b  2a  2a 1a 1a 1a 1a 1a 1a 1a
    11P 1a 3b 3a 7a 7b 2a  6b  6a 14a 14b 7c 7d 7e 7f 7g 7h 7i
    13P 1a 3a 3b 7b 7a 2a  6a  6b 14b 14a 7h 7i 7f 7e 7g 7c 7d

X.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
X.3      1  A /A  1  1 -1  -A -/A  -1  -1  1  1  1  1  1  1  1
X.4      1 /A  A  1  1 -1 -/A  -A  -1  -1  1  1  1  1  1  1  1
X.5      1  A /A  1  1  1   A  /A   1   1  1  1  1  1  1  1  1
X.6      1 /A  A  1  1  1  /A   A   1   1  1  1  1  1  1  1  1
X.7      3  .  .  B /B -3   .   .  -B -/B  B /B  B /B  3 /B  B
X.8      3  .  . /B  B -3   .   . -/B  -B /B  B /B  B  3  B /B
X.9      3  .  .  B /B  3   .   .   B  /B  B /B  B /B  3 /B  B
X.10     3  .  . /B  B  3   .   .  /B   B /B  B /B  B  3  B /B
X.11     6  .  . -1 -1  .   .   .   .   .  6 -1 -1 -1 -1  6 -1
X.12     6  .  . -1 -1  .   .   .   .   . /C /D  C /C -1  C  D
X.13     6  .  . -1 -1  .   .   .   .   .  C  D /C  C -1 /C /D
X.14     6  .  .  C /C  .   .   .   .   . /C -1  D /D -1  C -1
X.15     6  .  . /C  C  .   .   .   .   .  C -1 /D  D -1 /C -1
X.16     6  .  .  D /D  .   .   .   .   . /C /C -1 -1 -1  C  C
X.17     6  .  . /D  D  .   .   .   .   .  C  C -1 -1 -1 /C /C

A = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3
B = E(7)^3+E(7)^5+E(7)^6
  = (-1-Sqrt(-7))/2 = -1-b7
C = 2*E(7)+2*E(7)^2+2*E(7)^4
  = -1+Sqrt(-7) = 2b7
D = -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