Properties

Label 14T27
Order \(896\)
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 $t$ :  $27$
CHM label :  $2^{7}[1/2]D(7)$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (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:  $C_2$
14:  $D_{7}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 7: $D_{7}$

Low degree siblings

14T27 x 6, 14T28 x 7, 16T1078, 28T98, 28T105 x 7, 28T106 x 21, 28T107 x 21, 28T108 x 7, 28T109 x 7

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 $ $7$ $2$ $( 2, 9)( 5,12)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $7$ $2$ $( 2, 9)( 4,11)( 5,12)( 7,14)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $7$ $2$ $( 3,10)( 4,11)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $7$ $2$ $( 2, 9)( 3,10)( 4,11)( 5,12)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $14$ $2$ $( 3,10)( 5,12)( 6,13)( 7,14)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $7$ $2$ $( 2, 9)( 3,10)( 6,13)( 7,14)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $7$ $2$ $( 1, 8)( 6,13)$
$ 2, 2, 2, 2, 2, 2, 1, 1 $ $7$ $2$ $( 1, 8)( 2, 9)( 4,11)( 5,12)( 6,13)( 7,14)$
$ 7, 7 $ $128$ $7$ $( 1,14, 6, 5,11,10, 2)( 3, 9, 8, 7,13,12, 4)$
$ 7, 7 $ $128$ $7$ $( 1, 5, 2, 6,10,14,11)( 3, 7, 4, 8,12, 9,13)$
$ 7, 7 $ $128$ $7$ $( 1, 6,11, 2,14, 5,10)( 3, 8,13, 4, 9, 7,12)$
$ 4, 2, 2, 2, 2, 1, 1 $ $56$ $4$ $( 1, 3)( 4,14)( 5,13,12, 6)( 7,11)( 8,10)$
$ 2, 2, 2, 2, 2, 2, 2 $ $56$ $2$ $( 1, 3)( 2, 9)( 4,14)( 5,13)( 6,12)( 7,11)( 8,10)$
$ 4, 4, 4, 1, 1 $ $56$ $4$ $( 1,10, 8, 3)( 4, 7,11,14)( 5,13,12, 6)$
$ 4, 4, 2, 2, 2 $ $56$ $4$ $( 1,10, 8, 3)( 2, 9)( 4, 7,11,14)( 5,13)( 6,12)$
$ 4, 2, 2, 2, 2, 1, 1 $ $56$ $4$ $( 1, 3, 8,10)( 4,14)( 5, 6)( 7,11)(12,13)$
$ 4, 4, 2, 2, 2 $ $56$ $4$ $( 1, 3, 8,10)( 2, 9)( 4,14)( 5, 6,12,13)( 7,11)$
$ 4, 2, 2, 2, 2, 1, 1 $ $56$ $4$ $( 1,10)( 3, 8)( 4, 7,11,14)( 5, 6)(12,13)$
$ 4, 4, 2, 2, 2 $ $56$ $4$ $( 1,10)( 2, 9)( 3, 8)( 4, 7,11,14)( 5, 6,12,13)$

Group invariants

Order:  $896=2^{7} \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [896, 19344]
Character table:   
      2  7  7  7  7  7  6  7  7  7  .  .  .  4  4  4  4  4  4  4  4
      7  1  .  .  .  .  .  .  .  .  1  1  1  .  .  .  .  .  .  .  .

        1a 2a 2b 2c 2d 2e 2f 2g 2h 7a 7b 7c 4a 2i 4b 4c 4d 4e 4f 4g
     2P 1a 1a 1a 1a 1a 1a 1a 1a 1a 7c 7a 7b 2c 1a 2h 2f 2g 2b 2a 2d
     3P 1a 2a 2b 2c 2d 2e 2f 2g 2h 7b 7c 7a 4a 2i 4b 4c 4d 4e 4f 4g
     5P 1a 2a 2b 2c 2d 2e 2f 2g 2h 7c 7a 7b 4a 2i 4b 4c 4d 4e 4f 4g
     7P 1a 2a 2b 2c 2d 2e 2f 2g 2h 1a 1a 1a 4a 2i 4b 4c 4d 4e 4f 4g

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

A = E(7)+E(7)^6
B = E(7)^2+E(7)^5
C = E(7)^3+E(7)^4