Properties

Label 14T11
Order \(168\)
n \(14\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $F_8:C_3$

Related objects

Learn more about

Group action invariants

Degree $n$ :  $14$
Transitive number $t$ :  $11$
Group :  $F_8:C_3$
CHM label :  $[2^{3}]F_{21}(7)$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,9,11)(2,4,8)(3,13,5)(6,12,10), (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)
3:  $C_3$
21:  $C_7:C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 7: $C_7:C_3$

Low degree siblings

8T36, 24T283, 28T27, 42T26

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 $ $7$ $2$ $( 3,10)( 5,12)( 6,13)( 7,14)$
$ 3, 3, 3, 3, 1, 1 $ $28$ $3$ $( 2, 3, 5)( 4, 7,13)( 6,11,14)( 9,10,12)$
$ 3, 3, 3, 3, 1, 1 $ $28$ $3$ $( 2, 5, 3)( 4,13, 7)( 6,14,11)( 9,12,10)$
$ 7, 7 $ $24$ $7$ $( 1, 2, 3, 4,12, 6, 7)( 5,13,14, 8, 9,10,11)$
$ 6, 3, 3, 2 $ $28$ $6$ $( 1, 2, 4)( 3,13,12,10, 6, 5)( 7,14)( 8, 9,11)$
$ 6, 3, 3, 2 $ $28$ $6$ $( 1, 2, 6, 8, 9,13)( 3,10)( 4, 7, 5)(11,14,12)$
$ 7, 7 $ $24$ $7$ $( 1, 4,14,10,13, 2, 5)( 3, 6, 9,12, 8,11, 7)$

Group invariants

Order:  $168=2^{3} \cdot 3 \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [168, 43]
Character table:   
     2  3  3  1  1  .   1   1  .
     3  1  1  1  1  .   1   1  .
     7  1  .  .  .  1   .   .  1

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

X.1     1  1  1  1  1   1   1  1
X.2     1  1  A /A  1   A  /A  1
X.3     1  1 /A  A  1  /A   A  1
X.4     3  3  .  .  B   .   . /B
X.5     3  3  .  . /B   .   .  B
X.6     7 -1  1  1  .  -1  -1  .
X.7     7 -1  A /A  .  -A -/A  .
X.8     7 -1 /A  A  . -/A  -A  .

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