Properties

Label 14T31
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 $t$ :  $31$
CHM label :  $[D(7)^{2}:3_{3}]2$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (2,4,6,8,10,12,14), (2,12)(4,10)(6,8), (1,8)(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:  $C_2$ x 3
4:  $C_2^2$
6:  $S_3$
8:  $D_{4}$
12:  $D_{6}$
24:  $(C_6\times C_2):C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 7: None

Low degree siblings

28T133, 28T134, 28T135, 42T194, 42T196

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 $ $12$ $7$ $( 1, 3, 5, 7, 9,11,13)( 2, 8,14, 6,12, 4,10)$
$ 7, 7 $ $12$ $7$ $( 1, 3, 5, 7, 9,11,13)( 2, 6,10,14, 4, 8,12)$
$ 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 $ $98$ $3$ $( 3, 5, 9)( 4,10, 6)( 7,13,11)( 8,12,14)$
$ 6, 3, 3, 1, 1 $ $98$ $6$ $( 3, 5, 9)( 4, 8, 6,14,10,12)( 7,13,11)$
$ 6, 3, 3, 1, 1 $ $98$ $6$ $( 3,11, 9,13, 5, 7)( 4,10, 6)( 8,12,14)$
$ 6, 6, 1, 1 $ $98$ $6$ $( 3,11, 9,13, 5, 7)( 4, 8, 6,14,10,12)$
$ 2, 2, 2, 2, 2, 2, 2 $ $42$ $2$ $( 1, 8)( 2, 9)( 3,10)( 4,11)( 5,12)( 6,13)( 7,14)$
$ 14 $ $84$ $14$ $( 1,10, 3,12, 5,14, 7, 2, 9, 4,11, 6,13, 8)$
$ 14 $ $84$ $14$ $( 1,14, 7, 6,13,12, 5, 4,11,10, 3, 2, 9, 8)$
$ 14 $ $84$ $14$ $( 1,12, 5, 2, 9, 6,13,10, 3,14, 7, 4,11, 8)$
$ 4, 4, 4, 2 $ $294$ $4$ $( 1,10, 3, 8)( 2, 9)( 4,11,14, 7)( 5, 6,13,12)$

Group invariants

Order:  $1176=2^{3} \cdot 3 \cdot 7^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [1176, 216]
Character table:   
      2  3  1  1  1  1  2   1  3  2  2  2  2  2   1   1   1  2
      3  1  .  .  .  .  1   .  1  1  1  1  1  .   .   .   .  .
      7  2  2  2  2  2  1   1  .  .  .  .  .  1   1   1   1  .

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

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  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
X.5      2  2  2  2  2 -2  -2  2 -1  1  1 -1  .   .   .   .  .
X.6      2  2  2  2  2  2   2  2 -1 -1 -1 -1  .   .   .   .  .
X.7      2  2  2  2  2  .   . -2  2  .  . -2  .   .   .   .  .
X.8      2  2  2  2  2  .   . -2 -1  D -D  1  .   .   .   .  .
X.9      2  2  2  2  2  .   . -2 -1 -D  D  1  .   .   .   .  .
X.10    12  5 -2 -2 -2 -6   1  .  .  .  .  .  .   .   .   .  .
X.11    12  5 -2 -2 -2  6  -1  .  .  .  .  .  .   .   .   .  .
X.12    12 -2  A  C  B  .   .  .  .  .  .  . -2   E   F   G  .
X.13    12 -2  B  A  C  .   .  .  .  .  .  . -2   F   G   E  .
X.14    12 -2  C  B  A  .   .  .  .  .  .  . -2   G   E   F  .
X.15    12 -2  A  C  B  .   .  .  .  .  .  .  2  -E  -F  -G  .
X.16    12 -2  B  A  C  .   .  .  .  .  .  .  2  -F  -G  -E  .
X.17    12 -2  C  B  A  .   .  .  .  .  .  .  2  -G  -E  -F  .

A = -2*E(7)+E(7)^3+E(7)^4-2*E(7)^6
B = E(7)^2-2*E(7)^3-2*E(7)^4+E(7)^5
C = E(7)-2*E(7)^2-2*E(7)^5+E(7)^6
D = -E(3)+E(3)^2
  = -Sqrt(-3) = -i3
E = -E(7)^2-E(7)^5
F = -E(7)-E(7)^6
G = -E(7)^3-E(7)^4