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Properties

Label 14T19
Order \(336\)
n \(14\)
Cyclic No
Abelian No
Solvable No
Primitive No
$p$-group No
Group: $\GL(3,2) \times C_2$

Related objects

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Group action invariants

Degree $n$ :  $14$
Transitive number $t$ :  $19$
Group :  $\GL(3,2) \times C_2$
CHM label :  $L(7)[x]2$
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), (2,4)(5,13)(6,12)(9,11), (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:  $C_2$
168:  $\GL(3,2)$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 7: $\GL(3,2)$

Low degree siblings

14T17 x 2, 14T19, 16T714, 28T43 x 2, 42T78, 42T79, 42T80 x 2

Siblings are shown with degree $\leq 47$

A number field with this Galois group has exactly one arithmetically equivalent field.

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 $ $21$ $2$ $( 3, 5)( 6,14)( 7,13)(10,12)$
$ 4, 4, 2, 2, 1, 1 $ $42$ $4$ $( 2, 4)( 3, 5, 7,13)( 6,10,12,14)( 9,11)$
$ 3, 3, 3, 3, 1, 1 $ $56$ $3$ $( 2, 6,10)( 3, 9,13)( 4,12,14)( 5, 7,11)$
$ 2, 2, 2, 2, 2, 2, 2 $ $21$ $2$ $( 1, 2)( 3, 6)( 4,11)( 5,12)( 7,14)( 8, 9)(10,13)$
$ 4, 4, 2, 2, 2 $ $42$ $4$ $( 1, 2)( 3,12,13,14)( 4,11)( 5, 6, 7,10)( 8, 9)$
$ 14 $ $24$ $14$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14)$
$ 6, 6, 2 $ $56$ $6$ $( 1, 2, 3, 8, 9,10)( 4, 5,14,11,12, 7)( 6,13)$
$ 14 $ $24$ $14$ $( 1, 2, 3,14,13, 4, 5, 8, 9,10, 7, 6,11,12)$
$ 7, 7 $ $24$ $7$ $( 1, 3, 5, 7, 9,11,13)( 2, 4, 6, 8,10,12,14)$
$ 7, 7 $ $24$ $7$ $( 1, 3, 9,11,13, 7, 5)( 2, 4, 6,14,12, 8,10)$
$ 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:  No
GAP id:  [336, 209]
Character table:    
      2  4  4  3  1  4  3   1  1   1  1  1  4
      3  1  .  .  1  .  .   .  1   .  .  .  1
      7  1  .  .  .  .  .   1  .   1  1  1  1

        1a 2a 4a 3a 2b 4b 14a 6a 14b 7a 7b 2c
     2P 1a 1a 2a 3a 1a 2a  7a 3a  7b 7a 7b 1a
     3P 1a 2a 4a 1a 2b 4b 14b 2c 14a 7b 7a 2c
     5P 1a 2a 4a 3a 2b 4b 14b 6a 14a 7b 7a 2c
     7P 1a 2a 4a 3a 2b 4b  2c 6a  2c 1a 1a 2c
    11P 1a 2a 4a 3a 2b 4b 14a 6a 14b 7a 7b 2c
    13P 1a 2a 4a 3a 2b 4b 14b 6a 14a 7b 7a 2c

X.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
X.3      3 -1  1  . -1  1   A  .  /A  A /A  3
X.4      3 -1  1  . -1  1  /A  .   A /A  A  3
X.5      3 -1  1  .  1 -1 -/A  .  -A /A  A -3
X.6      3 -1  1  .  1 -1  -A  . -/A  A /A -3
X.7      6  2  .  .  2  .  -1  .  -1 -1 -1  6
X.8      6  2  .  . -2  .   1  .   1 -1 -1 -6
X.9      7 -1 -1  1 -1 -1   .  1   .  .  .  7
X.10     7 -1 -1  1  1  1   . -1   .  .  . -7
X.11     8  .  . -1  .  .   1 -1   1  1  1  8
X.12     8  .  . -1  .  .  -1  1  -1  1  1 -8

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