Properties

Label 14T35
Order \(1344\)
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 $t$ :  $35$
CHM label :  $[2^{6}]F_{21}(7)$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,9,11)(2,4,8)(3,13,5)(6,12,10), (2,9)(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)
3:  $C_3$
21:  $C_7:C_3$
168:  $C_2^3:(C_7: C_3)$ x 2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 7: $C_7:C_3$

Low degree siblings

28T154, 28T155 x 2, 28T157, 42T202, 42T204, 42T205

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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $21$ $2$ $( 1, 8)( 2, 9)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $21$ $2$ $( 1, 8)( 2, 9)( 3,10)( 4,11)$
$ 2, 2, 2, 2, 2, 2, 1, 1 $ $7$ $2$ $( 1, 8)( 2, 9)( 3,10)( 4,11)( 5,12)( 6,13)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $7$ $2$ $( 1, 8)( 2, 9)( 3,10)( 6,13)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $7$ $2$ $( 1, 8)( 2, 9)( 3,10)( 5,12)$
$ 7, 7 $ $192$ $7$ $( 1,13, 4, 9, 7,12,10)( 2,14, 5, 3, 8, 6,11)$
$ 7, 7 $ $192$ $7$ $( 1, 9,10, 4,12,13, 7)( 2, 3,11, 5, 6,14, 8)$
$ 3, 3, 3, 3, 1, 1 $ $112$ $3$ $( 2,10, 5)( 3,12, 9)( 4,14, 6)( 7,13,11)$
$ 6, 3, 3, 2 $ $112$ $6$ $( 1, 8)( 2,10, 5, 9, 3,12)( 4,14, 6)( 7,13,11)$
$ 6, 6, 1, 1 $ $112$ $6$ $( 2, 3,12, 9,10, 5)( 4,14, 6,11, 7,13)$
$ 6, 3, 3, 2 $ $112$ $6$ $( 1, 8)( 2, 3,12)( 4,14, 6,11, 7,13)( 5, 9,10)$
$ 3, 3, 3, 3, 1, 1 $ $112$ $3$ $( 2, 5,10)( 3, 9,12)( 4, 6,14)( 7,11,13)$
$ 6, 3, 3, 2 $ $112$ $6$ $( 1, 8)( 2, 5,10, 9,12, 3)( 4, 6,14)( 7,11,13)$
$ 6, 6, 1, 1 $ $112$ $6$ $( 2, 5, 3, 9,12,10)( 4, 6,14,11,13, 7)$
$ 6, 3, 3, 2 $ $112$ $6$ $( 1, 8)( 2, 5, 3)( 4, 6,14,11,13, 7)( 9,12,10)$

Group invariants

Order:  $1344=2^{6} \cdot 3 \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [1344, 11691]
Character table:   
      2  6  6  6  6  6  6  .  .  2   2   2   2  2   2   2   2
      3  1  .  .  1  1  1  .  .  1   1   1   1  1   1   1   1
      7  1  .  .  .  .  .  1  1  .   .   .   .  .   .   .   .

        1a 2a 2b 2c 2d 2e 7a 7b 3a  6a  6b  6c 3b  6d  6e  6f
     2P 1a 1a 1a 1a 1a 1a 7a 7b 3b  3b  3b  3b 3a  3a  3a  3a
     3P 1a 2a 2b 2c 2d 2e 7b 7a 1a  2e  2c  2d 1a  2e  2c  2d
     5P 1a 2a 2b 2c 2d 2e 7b 7a 3b  6d  6e  6f 3a  6a  6b  6c
     7P 1a 2a 2b 2c 2d 2e 1a 1a 3a  6a  6b  6c 3b  6d  6e  6f

X.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  B   B   B   B /B  /B  /B  /B
X.3      1  1  1  1  1  1  1  1 /B  /B  /B  /B  B   B   B   B
X.4      3  3  3  3  3  3  A /A  .   .   .   .  .   .   .   .
X.5      3  3  3  3  3  3 /A  A  .   .   .   .  .   .   .   .
X.6      7 -1 -1 -1 -1  7  .  .  1   1  -1  -1  1   1  -1  -1
X.7      7 -1 -1 -1  7 -1  .  .  1  -1  -1   1  1  -1  -1   1
X.8      7  3 -1 -5 -1 -1  .  .  1  -1   1  -1  1  -1   1  -1
X.9      7 -1 -1 -1 -1  7  .  .  B   B  -B  -B /B  /B -/B -/B
X.10     7 -1 -1 -1 -1  7  .  . /B  /B -/B -/B  B   B  -B  -B
X.11     7 -1 -1 -1  7 -1  .  .  B  -B  -B   B /B -/B -/B  /B
X.12     7 -1 -1 -1  7 -1  .  . /B -/B -/B  /B  B  -B  -B   B
X.13     7  3 -1 -5 -1 -1  .  .  B  -B   B  -B /B -/B  /B -/B
X.14     7  3 -1 -5 -1 -1  .  . /B -/B  /B -/B  B  -B   B  -B
X.15    21  1 -3  9 -3 -3  .  .  .   .   .   .  .   .   .   .
X.16    21 -3  5 -3 -3 -3  .  .  .   .   .   .  .   .   .   .

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