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Properties

Label 14T40
Order \(2688\)
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 $t$ :  $40$
CHM label :  $1/2[2^{7}]F_{42}(7)$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,9,11)(2,4,8)(3,13,5)(6,12,10), (1,13)(2,12)(3,11)(4,10)(5,9)(6,8)(7,14), (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)
2:  $C_2$
3:  $C_3$
6:  $C_6$
42:  $F_7$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 7: $F_7$

Low degree siblings

14T41, 16T1502, 28T215, 28T227, 28T228, 28T237, 42T314, 42T315, 42T316, 42T317, 42T318, 42T319

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 $ $21$ $2$ $( 1, 8)( 2, 9)( 5,12)( 6,13)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $14$ $2$ $( 1, 8)( 2, 9)( 3,10)( 6,13)$
$ 2, 2, 2, 2, 2, 2, 1, 1 $ $7$ $2$ $( 1, 8)( 2, 9)( 3,10)( 5,12)( 6,13)( 7,14)$
$ 7, 7 $ $384$ $7$ $( 1,13, 4, 9, 7,12,10)( 2,14, 5, 3, 8, 6,11)$
$ 3, 3, 3, 3, 1, 1 $ $112$ $3$ $( 2,10, 5)( 3,12, 9)( 4,14, 6)( 7,13,11)$
$ 6, 3, 3, 2 $ $224$ $6$ $( 1, 8)( 2,10, 5, 9, 3,12)( 4,14, 6)( 7,13,11)$
$ 6, 6, 1, 1 $ $112$ $6$ $( 2,10,12, 9, 3, 5)( 4,14,13,11, 7, 6)$
$ 3, 3, 3, 3, 1, 1 $ $112$ $3$ $( 2, 5,10)( 3, 9,12)( 4, 6,14)( 7,11,13)$
$ 6, 3, 3, 2 $ $224$ $6$ $( 1, 8)( 2, 5,10, 9,12, 3)( 4, 6,14)( 7,11,13)$
$ 6, 6, 1, 1 $ $112$ $6$ $( 2,12, 3, 9, 5,10)( 4,13, 7,11, 6,14)$
$ 4, 2, 2, 2, 2, 1, 1 $ $168$ $4$ $( 2, 7)( 3, 6)( 4,12,11, 5)( 9,14)(10,13)$
$ 4, 4, 2, 2, 2 $ $168$ $4$ $( 1, 8)( 2, 7, 9,14)( 3, 6)( 4,12,11, 5)(10,13)$
$ 4, 4, 4, 1, 1 $ $56$ $4$ $( 2,14, 9, 7)( 3,13,10, 6)( 4,12,11, 5)$
$ 2, 2, 2, 2, 2, 2, 2 $ $56$ $2$ $( 1, 8)( 2,14)( 3, 6)( 4, 5)( 7, 9)(10,13)(11,12)$
$ 12, 1, 1 $ $224$ $12$ $( 2,13, 5,14, 3, 4, 9, 6,12, 7,10,11)$
$ 6, 6, 2 $ $224$ $6$ $( 1, 8)( 2,13, 5,14, 3, 4)( 6,12, 7,10,11, 9)$
$ 12, 1, 1 $ $224$ $12$ $( 2,11,10, 7, 5, 6, 9, 4, 3,14,12,13)$
$ 6, 6, 2 $ $224$ $6$ $( 1, 8)( 2,11,10, 7, 5, 6)( 3,14,12,13, 9, 4)$

Group invariants

Order:  $2688=2^{7} \cdot 3 \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  Data not available
Character table:    
      2  7  7  7  6  7  .  3   2   3  3   2   3  4  4  4  4   2   2   2   2
      3  1  .  .  1  1  .  1   1   1  1   1   1  .  .  1  1   1   1   1   1
      7  1  .  .  .  .  1  .   .   .  .   .   .  .  .  .  .   .   .   .   .

        1a 2a 2b 2c 2d 7a 3a  6a  6b 3b  6c  6d 4a 4b 4c 2e 12a  6e 12b  6f
     2P 1a 1a 1a 1a 1a 7a 3b  3b  3b 3a  3a  3a 2a 2b 2d 1a  6d  3b  6b  3a
     3P 1a 2a 2b 2c 2d 7a 1a  2c  2d 1a  2c  2d 4a 4b 4c 2e  4c  2e  4c  2e
     5P 1a 2a 2b 2c 2d 7a 3b  6c  6d 3a  6a  6b 4a 4b 4c 2e 12b  6f 12a  6e
     7P 1a 2a 2b 2c 2d 1a 3a  6a  6b 3b  6c  6d 4a 4b 4c 2e 12a  6e 12b  6f
    11P 1a 2a 2b 2c 2d 7a 3b  6c  6d 3a  6a  6b 4a 4b 4c 2e 12b  6f 12a  6e

X.1      1  1  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  -1  -1  -1
X.3      1  1  1  1  1  1  A   A   A /A  /A  /A -1 -1 -1 -1  -A  -A -/A -/A
X.4      1  1  1  1  1  1 /A  /A  /A  A   A   A -1 -1 -1 -1 -/A -/A  -A  -A
X.5      1  1  1  1  1  1  A   A   A /A  /A  /A  1  1  1  1   A   A  /A  /A
X.6      1  1  1  1  1  1 /A  /A  /A  A   A   A  1  1  1  1  /A  /A   A   A
X.7      6  6  6  6  6 -1  .   .   .  .   .   .  .  .  .  .   .   .   .   .
X.8      7  3 -1 -1 -5  .  1  -1   1  1  -1   1  1 -1  1 -1   1  -1   1  -1
X.9      7  3 -1 -1 -5  .  1  -1   1  1  -1   1 -1  1 -1  1  -1   1  -1   1
X.10     7  3 -1 -1 -5  . /A -/A  /A  A  -A   A  1 -1  1 -1  /A -/A   A  -A
X.11     7  3 -1 -1 -5  .  A  -A   A /A -/A  /A  1 -1  1 -1   A  -A  /A -/A
X.12     7  3 -1 -1 -5  . /A -/A  /A  A  -A   A -1  1 -1  1 -/A  /A  -A   A
X.13     7  3 -1 -1 -5  .  A  -A   A /A -/A  /A -1  1 -1  1  -A   A -/A  /A
X.14    14 -2 -2  6 -2  .  2   .  -2  2   .  -2  .  .  .  .   .   .   .   .
X.15    14 -2 -2  6 -2  .  B   .  -B /B   . -/B  .  .  .  .   .   .   .   .
X.16    14 -2 -2  6 -2  . /B   . -/B  B   .  -B  .  .  .  .   .   .   .   .
X.17    21  1 -3 -3  9  .  .   .   .  .   .   . -1  1  3 -3   .   .   .   .
X.18    21  1 -3 -3  9  .  .   .   .  .   .   .  1 -1 -3  3   .   .   .   .
X.19    21 -3  5 -3 -3  .  .   .   .  .   .   . -1 -1  3  3   .   .   .   .
X.20    21 -3  5 -3 -3  .  .   .   .  .   .   .  1  1 -3 -3   .   .   .   .

A = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3
B = 2*E(3)
  = -1+Sqrt(-3) = 2b3