Properties

Label 14T7
Order \(84\)
n \(14\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $F_7 \times C_2$

Related objects

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Group action invariants

Degree $n$ :  $14$
Transitive number $t$ :  $7$
Group :  $F_7 \times C_2$
CHM label :  $F_{42}(7)[x]2$
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)(3,10)(4,11)(5,12)(6,13)(7,14), (1,13)(2,12)(3,11)(4,10)(5,9)(6,8), (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:  $C_2$ x 3
3:  $C_3$
4:  $C_2^2$
6:  $C_6$ x 3
12:  $C_6\times C_2$
42:  $F_7$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 7: $F_7$

Low degree siblings

14T7, 28T15, 42T10 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 $ $1$ $1$ $()$
$ 6, 6, 1, 1 $ $7$ $6$ $( 2, 4,10,14,12, 6)( 3, 7, 5,13, 9,11)$
$ 6, 6, 1, 1 $ $7$ $6$ $( 2, 6,12,14,10, 4)( 3,11, 9,13, 5, 7)$
$ 3, 3, 3, 3, 1, 1 $ $7$ $3$ $( 2,10,12)( 3, 5, 9)( 4,14, 6)( 7,13,11)$
$ 3, 3, 3, 3, 1, 1 $ $7$ $3$ $( 2,12,10)( 3, 9, 5)( 4, 6,14)( 7,11,13)$
$ 2, 2, 2, 2, 2, 2, 1, 1 $ $7$ $2$ $( 2,14)( 3,13)( 4,12)( 5,11)( 6,10)( 7, 9)$
$ 2, 2, 2, 2, 2, 2, 2 $ $7$ $2$ $( 1, 2)( 3,14)( 4,13)( 5,12)( 6,11)( 7,10)( 8, 9)$
$ 14 $ $6$ $14$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14)$
$ 6, 6, 2 $ $7$ $6$ $( 1, 2, 5,14,13,10)( 3, 8, 9,12, 7, 6)( 4,11)$
$ 6, 6, 2 $ $7$ $6$ $( 1, 2, 7, 4, 3,12)( 5, 8, 9,14,11,10)( 6,13)$
$ 6, 6, 2 $ $7$ $6$ $( 1, 2,11, 8, 9, 4)( 3, 6, 5,10,13,12)( 7,14)$
$ 6, 6, 2 $ $7$ $6$ $( 1, 2,13, 8, 9, 6)( 3,10)( 4, 7,12,11,14, 5)$
$ 7, 7 $ $6$ $7$ $( 1, 3, 5, 7, 9,11,13)( 2, 4, 6, 8,10,12,14)$
$ 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:  $84=2^{2} \cdot 3 \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [84, 7]
Character table:   
      2  2   2   2   2   2  2  2   1   2   2   2   2  1  2
      3  1   1   1   1   1  1  1   .   1   1   1   1  .  1
      7  1   .   .   .   .  .  .   1   .   .   .   .  1  1

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

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

A = -E(3)
  = (1-Sqrt(-3))/2 = -b3