Properties

Label 14T32
Order \(1176\)
n \(14\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No

Related objects

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Group action invariants

Degree $n$ :  $14$
Transitive number $t$ :  $32$
CHM label :  $[D(7)^{2}:3]2$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (2,4,6,8,10,12,14), (2,12)(4,10)(6,8), (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:  $C_2$ x 3
3:  $C_3$
4:  $C_2^2$
6:  $C_6$ x 3
8:  $D_{4}$
12:  $C_6\times C_2$
24:  $D_4 \times C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 7: None

Low degree siblings

14T32, 28T136 x 2, 28T137 x 2, 28T138 x 2, 28T143, 42T195 x 2

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 $ $1$ $1$ $()$
$ 7, 1, 1, 1, 1, 1, 1, 1 $ $12$ $7$ $( 2, 4, 6, 8,10,12,14)$
$ 7, 7 $ $12$ $7$ $( 1, 3, 5, 7, 9,11,13)( 2, 4, 6, 8,10,12,14)$
$ 7, 7 $ $24$ $7$ $( 1, 3, 5, 7, 9,11,13)( 2, 8,14, 6,12, 4,10)$
$ 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $14$ $2$ $( 4,14)( 6,12)( 8,10)$
$ 7, 2, 2, 2, 1 $ $84$ $14$ $( 1, 3, 5, 7, 9,11,13)( 4,14)( 6,12)( 8,10)$
$ 2, 2, 2, 2, 2, 2, 1, 1 $ $49$ $2$ $( 3,13)( 4,14)( 5,11)( 6,12)( 7, 9)( 8,10)$
$ 3, 3, 3, 3, 1, 1 $ $49$ $3$ $( 3, 5, 9)( 4, 6,10)( 7,13,11)( 8,14,12)$
$ 6, 3, 3, 1, 1 $ $98$ $6$ $( 3, 5, 9)( 4,12,10,14, 6, 8)( 7,13,11)$
$ 6, 6, 1, 1 $ $49$ $6$ $( 3,11, 9,13, 5, 7)( 4,12,10,14, 6, 8)$
$ 3, 3, 3, 3, 1, 1 $ $49$ $3$ $( 3, 9, 5)( 4,10, 6)( 7,11,13)( 8,12,14)$
$ 6, 3, 3, 1, 1 $ $98$ $6$ $( 3, 9, 5)( 4, 8, 6,14,10,12)( 7,11,13)$
$ 6, 6, 1, 1 $ $49$ $6$ $( 3, 7, 5,13, 9,11)( 4, 8, 6,14,10,12)$
$ 2, 2, 2, 2, 2, 2, 2 $ $14$ $2$ $( 1, 8)( 2, 9)( 3,10)( 4,11)( 5,12)( 6,13)( 7,14)$
$ 14 $ $84$ $14$ $( 1,10, 3,12, 5,14, 7, 2, 9, 4,11, 6,13, 8)$
$ 4, 4, 4, 2 $ $98$ $4$ $( 1,10, 3, 8)( 2, 9)( 4,11,14, 7)( 5, 6,13,12)$
$ 6, 6, 2 $ $98$ $6$ $( 1,14,13,10, 5, 8)( 2, 3, 4, 7,12, 9)( 6,11)$
$ 12, 2 $ $98$ $12$ $( 1, 4, 7, 6,11,12, 9, 2, 3,14,13, 8)( 5,10)$
$ 6, 6, 2 $ $98$ $6$ $( 1,12, 3, 6, 7, 8)( 2, 5,14,11,10, 9)( 4,13)$
$ 12, 2 $ $98$ $12$ $( 1, 6, 7,10, 9, 2, 5, 4,13,14,11, 8)( 3,12)$

Group invariants

Order:  $1176=2^{3} \cdot 3 \cdot 7^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [1176, 217]
Character table:   
      2  3  1  1  .  2   1  3  3   2   3  3   2   3  2   1  2   2   2   2   2
      3  1  .  .  .  1   .  1  1   1   1  1   1   1  1   .  1   1   1   1   1
      7  2  2  2  2  1   1  .  .   .   .  .   .   .  1   1  .   .   .   .   .

        1a 7a 7b 7c 2a 14a 2b 3a  6a  6b 3b  6c  6d 2c 14b 4a  6e 12a  6f 12b
     2P 1a 7a 7b 7c 1a  7a 1a 3b  3b  3b 3a  3a  3a 1a  7b 2b  3b  6d  3a  6b
     3P 1a 7a 7b 7c 2a 14a 2b 1a  2a  2b 1a  2a  2b 2c 14b 4a  2c  4a  2c  4a
     5P 1a 7a 7b 7c 2a 14a 2b 3b  6c  6d 3a  6a  6b 2c 14b 4a  6f 12b  6e 12a
     7P 1a 1a 1a 1a 2a  2a 2b 3a  6a  6b 3b  6c  6d 2c  2c 4a  6e 12a  6f 12b
    11P 1a 7a 7b 7c 2a 14a 2b 3b  6c  6d 3a  6a  6b 2c 14b 4a  6f 12b  6e 12a
    13P 1a 7a 7b 7c 2a 14a 2b 3a  6a  6b 3b  6c  6d 2c 14b 4a  6e 12a  6f 12b

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      1  1  1  1 -1  -1  1  A  -A   A /A -/A  /A -1  -1  1  -A   A -/A  /A
X.6      1  1  1  1 -1  -1  1 /A -/A  /A  A  -A   A -1  -1  1 -/A  /A  -A   A
X.7      1  1  1  1 -1  -1  1  A  -A   A /A -/A  /A  1   1 -1   A  -A  /A -/A
X.8      1  1  1  1 -1  -1  1 /A -/A  /A  A  -A   A  1   1 -1  /A -/A   A  -A
X.9      1  1  1  1  1   1  1  A   A   A /A  /A  /A -1  -1 -1  -A  -A -/A -/A
X.10     1  1  1  1  1   1  1 /A  /A  /A  A   A   A -1  -1 -1 -/A -/A  -A  -A
X.11     1  1  1  1  1   1  1  A   A   A /A  /A  /A  1   1  1   A   A  /A  /A
X.12     1  1  1  1  1   1  1 /A  /A  /A  A   A   A  1   1  1  /A  /A   A   A
X.13     2  2  2  2  .   . -2  2   .  -2  2   .  -2  .   .  .   .   .   .   .
X.14     2  2  2  2  .   . -2  B   .  -B /B   . -/B  .   .  .   .   .   .   .
X.15     2  2  2  2  .   . -2 /B   . -/B  B   .  -B  .   .  .   .   .   .   .
X.16    12  5 -2 -2 -6   1  .  .   .   .  .   .   .  .   .  .   .   .   .   .
X.17    12  5 -2 -2  6  -1  .  .   .   .  .   .   .  .   .  .   .   .   .   .
X.18    12 -2  5 -2  .   .  .  .   .   .  .   .   . -6   1  .   .   .   .   .
X.19    12 -2  5 -2  .   .  .  .   .   .  .   .   .  6  -1  .   .   .   .   .
X.20    24 -4 -4  3  .   .  .  .   .   .  .   .   .  .   .  .   .   .   .   .

A = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3
B = 2*E(3)
  = -1+Sqrt(-3) = 2b3