Properties

Label 14T34
Order \(1344\)
n \(14\)
Cyclic No
Abelian No
Solvable No
Primitive No
$p$-group No

Related objects

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Group action invariants

Degree $n$ :  $14$
Transitive number $t$ :  $34$
CHM label :  $2^{3}:L_{7}(14)=[2^{3}]L(7)=[2^{3}]L(3,2)$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,9,11)(2,4,8)(3,13,5)(6,12,10), (2,4)(5,13)(6,12)(9,11), (3,10)(5,12)(6,13)(7,14), (1,3,5,7,9,11,13)(2,4,6,8,10,12,14)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
168:  $\GL(3,2)$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 7: $\GL(3,2)$

Low degree siblings

8T48 x 2, 14T34, 28T153, 28T159 x 2, 42T210 x 2, 42T211 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 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $7$ $2$ $( 3,10)( 5,12)( 6,13)( 7,14)$
$ 2, 2, 2, 2, 2, 2, 1, 1 $ $42$ $2$ $( 1, 8)( 3,12)( 4,11)( 5,10)( 6,14)( 7,13)$
$ 4, 4, 2, 2, 1, 1 $ $84$ $4$ $( 2, 9)( 3, 5,10,12)( 4,11)( 6, 7,13,14)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $42$ $2$ $( 3,12)( 5,10)( 6, 7)(13,14)$
$ 4, 4, 4, 2 $ $168$ $4$ $( 1, 8)( 2, 3,11, 7)( 4,14, 9,10)( 5, 6,12,13)$
$ 4, 4, 2, 2, 1, 1 $ $168$ $4$ $( 2, 3,11,14)( 4, 7, 9,10)( 5, 6)(12,13)$
$ 3, 3, 3, 3, 1, 1 $ $224$ $3$ $( 2, 3,12)( 4, 7, 6)( 5, 9,10)(11,14,13)$
$ 6, 3, 3, 2 $ $224$ $6$ $( 1, 8)( 2, 3, 5)( 4,14,13,11, 7, 6)( 9,10,12)$
$ 7, 7 $ $192$ $7$ $( 1, 2, 3, 4,12, 6, 7)( 5,13,14, 8, 9,10,11)$
$ 7, 7 $ $192$ $7$ $( 1, 9,10, 7, 6, 4, 5)( 2, 3,14,13,11,12, 8)$

Group invariants

Order:  $1344=2^{6} \cdot 3 \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  [1344, 11686]
Character table:   
      2  6  6  5  4  5  3  3  1  1  .  .
      3  1  1  .  .  .  .  .  1  1  .  .
      7  1  .  .  .  .  .  .  .  .  1  1

        1a 2a 2b 4a 2c 4b 4c 3a 6a 7a 7b
     2P 1a 1a 1a 2a 1a 2b 2c 3a 3a 7a 7b
     3P 1a 2a 2b 4a 2c 4b 4c 1a 2a 7b 7a
     5P 1a 2a 2b 4a 2c 4b 4c 3a 6a 7b 7a
     7P 1a 2a 2b 4a 2c 4b 4c 3a 6a 1a 1a

X.1      1  1  1  1  1  1  1  1  1  1  1
X.2      3  3 -1 -1 -1  1  1  .  .  A /A
X.3      3  3 -1 -1 -1  1  1  .  . /A  A
X.4      6  6  2  2  2  .  .  .  . -1 -1
X.5      7 -1  3 -1 -1  1 -1  1 -1  .  .
X.6      7  7 -1 -1 -1 -1 -1  1  1  .  .
X.7      7 -1 -1 -1  3 -1  1  1 -1  .  .
X.8      8  8  .  .  .  .  . -1 -1  1  1
X.9     14 -2  2 -2  2  .  . -1  1  .  .
X.10    21 -3  1  1 -3 -1  1  .  .  .  .
X.11    21 -3 -3  1  1  1 -1  .  .  .  .

A = E(7)^3+E(7)^5+E(7)^6
  = (-1-Sqrt(-7))/2 = -1-b7