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Properties

Label 14T25
Order \(588\)
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 $t$ :  $25$
CHM label :  $[7^{2}:6_{3}]2$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (2,4,6,8,10,12,14), (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), (1,11,9)(2,4,8)(3,5,13)(6,12,10)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
6:  $S_3$
12:  $D_{6}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 7: None

Low degree siblings

21T23 x 2, 28T78, 42T110 x 2, 42T111 x 2, 42T112 x 2, 42T122

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 $ $1$ $1$ $()$
$ 7, 1, 1, 1, 1, 1, 1, 1 $ $12$ $7$ $( 2, 4, 6, 8,10,12,14)$
$ 7, 7 $ $6$ $7$ $( 1, 3, 5, 7, 9,11,13)( 2, 4, 6, 8,10,12,14)$
$ 7, 7 $ $6$ $7$ $( 1, 3, 5, 7, 9,11,13)( 2, 8,14, 6,12, 4,10)$
$ 7, 7 $ $6$ $7$ $( 1, 3, 5, 7, 9,11,13)( 2, 6,10,14, 4, 8,12)$
$ 7, 7 $ $6$ $7$ $( 1, 3, 5, 7, 9,11,13)( 2,14,12,10, 8, 6, 4)$
$ 7, 7 $ $6$ $7$ $( 1, 3, 5, 7, 9,11,13)( 2,10, 4,12, 6,14, 8)$
$ 7, 7 $ $6$ $7$ $( 1, 3, 5, 7, 9,11,13)( 2,12, 8, 4,14,10, 6)$
$ 3, 3, 3, 3, 1, 1 $ $98$ $3$ $( 3, 5, 9)( 4,10, 6)( 7,13,11)( 8,12,14)$
$ 2, 2, 2, 2, 2, 2, 2 $ $21$ $2$ $( 1, 8)( 2, 9)( 3,10)( 4,11)( 5,12)( 6,13)( 7,14)$
$ 14 $ $42$ $14$ $( 1,10, 3,12, 5,14, 7, 2, 9, 4,11, 6,13, 8)$
$ 14 $ $42$ $14$ $( 1,14, 7, 6,13,12, 5, 4,11,10, 3, 2, 9, 8)$
$ 14 $ $42$ $14$ $( 1,12, 5, 2, 9, 6,13,10, 3,14, 7, 4,11, 8)$
$ 2, 2, 2, 2, 2, 2, 1, 1 $ $49$ $2$ $( 3,13)( 4,14)( 5,11)( 6,12)( 7, 9)( 8,10)$
$ 6, 6, 1, 1 $ $98$ $6$ $( 3,11, 9,13, 5, 7)( 4, 8, 6,14,10,12)$
$ 14 $ $42$ $14$ $( 1, 8, 3, 6, 5, 4, 7, 2, 9,14,11,12,13,10)$
$ 2, 2, 2, 2, 2, 2, 2 $ $21$ $2$ $( 1,10)( 2, 9)( 3, 8)( 4, 7)( 5, 6)(11,14)(12,13)$
$ 14 $ $42$ $14$ $( 1,14,11, 4, 7, 8, 3,12,13, 2, 9, 6, 5,10)$
$ 14 $ $42$ $14$ $( 1, 2, 9, 8, 3,14,11, 6, 5,12,13, 4, 7,10)$

Group invariants

Order:  $588=2^{2} \cdot 3 \cdot 7^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [588, 35]
Character table:    
      2  2  .  1  1  1  1  1  1  1  2   1   1   1  2  1   1  2   1   1
      3  1  .  .  .  .  .  .  .  1  .   .   .   .  1  1   .  .   .   .
      7  2  2  2  2  2  2  2  2  .  1   1   1   1  .  .   1  1   1   1

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

X.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
X.3      1  1  1  1  1  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 -1  -1 -1  -1  -1
X.5      2  2  2  2  2  2  2  2 -1  .   .   .   . -2  1   .  .   .   .
X.6      2  2  2  2  2  2  2  2 -1  .   .   .   .  2 -1   .  .   .   .
X.7      6 -1  A  E  B  D  C  F  .  .   .   .   .  .  .   G -2   H   I
X.8      6 -1  B  D  C  F  A  E  .  .   .   .   .  .  .   I -2   G   H
X.9      6 -1  C  F  A  E  B  D  .  .   .   .   .  .  .   H -2   I   G
X.10     6 -1  A  E  B  D  C  F  .  .   .   .   .  .  .  -G  2  -H  -I
X.11     6 -1  B  D  C  F  A  E  .  .   .   .   .  .  .  -I  2  -G  -H
X.12     6 -1  C  F  A  E  B  D  .  .   .   .   .  .  .  -H  2  -I  -G
X.13     6 -1  D  C  F  A  E  B  . -2   G   I   H  .  .   .  .   .   .
X.14     6 -1  E  B  D  C  F  A  . -2   H   G   I  .  .   .  .   .   .
X.15     6 -1  F  A  E  B  D  C  . -2   I   H   G  .  .   .  .   .   .
X.16     6 -1  D  C  F  A  E  B  .  2  -G  -I  -H  .  .   .  .   .   .
X.17     6 -1  E  B  D  C  F  A  .  2  -H  -G  -I  .  .   .  .   .   .
X.18     6 -1  F  A  E  B  D  C  .  2  -I  -H  -G  .  .   .  .   .   .
X.19    12  5 -2 -2 -2 -2 -2 -2  .  .   .   .   .  .  .   .  .   .   .

A = -2*E(7)-2*E(7)^2-2*E(7)^5-2*E(7)^6
B = -2*E(7)-2*E(7)^3-2*E(7)^4-2*E(7)^6
C = -2*E(7)^2-2*E(7)^3-2*E(7)^4-2*E(7)^5
D = 2*E(7)^2+E(7)^3+E(7)^4+2*E(7)^5
E = E(7)+2*E(7)^3+2*E(7)^4+E(7)^6
F = 2*E(7)+E(7)^2+E(7)^5+2*E(7)^6
G = -E(7)^2-E(7)^5
H = -E(7)^3-E(7)^4
I = -E(7)-E(7)^6