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Properties

Label 14T18
Order \(336\)
n \(14\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2\times F_8:C_3$

Related objects

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Group action invariants

Degree $n$ :  $14$
Transitive number $t$ :  $18$
Group :  $C_2\times F_8:C_3$
CHM label :  $[2^{4}]F_{21}(7)$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,9,11)(2,4,8)(3,13,5)(6,12,10), (1,8)(2,9)(4,11), (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)
2:  $C_2$
3:  $C_3$
6:  $C_6$
21:  $C_7:C_3$
42:  $(C_7:C_3) \times C_2$
168:  $C_2^3:(C_7: C_3)$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 7: $C_7:C_3$

Low degree siblings

16T712, 28T44, 42T67

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

Group invariants

Order:  $336=2^{4} \cdot 3 \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [336, 210]
Character table:    
      2  4  4  4  2   2   2  2   1   1   2   2   2   2   1   1  4
      3  1  1  1  1   1   1  1   .   .   1   1   1   1   .   .  1
      7  1  .  .  .   .   .  .   1   1   .   .   .   .   1   1  1

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

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

A = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3
B = -E(7)-E(7)^2-E(7)^4
  = (1-Sqrt(-7))/2 = -b7