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Properties

Label 14T26
Order \(882\)
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 $t$ :  $26$
CHM label :  $1/2[1/2.F_{42}(7)^{2}]2$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (2,4,6,8,10,12,14), (1,8)(2,5)(3,4)(6,13)(7,14)(9,12)(10,11), (2,4,8)(6,12,10)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
6:  $S_3$, $C_6$
18:  $S_3\times C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 7: None

Low degree siblings

21T25, 21T26, 42T143, 42T144, 42T152, 42T153, 42T154, 42T155

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 $ $6$ $7$ $( 2, 4, 6, 8,10,12,14)$
$ 7, 1, 1, 1, 1, 1, 1, 1 $ $6$ $7$ $( 2, 8,14, 6,12, 4,10)$
$ 7, 7 $ $9$ $7$ $( 1,11, 9, 7, 5, 3,13)( 2, 4, 6, 8,10,12,14)$
$ 7, 7 $ $18$ $7$ $( 1,11, 9, 7, 5, 3,13)( 2, 8,14, 6,12, 4,10)$
$ 7, 7 $ $9$ $7$ $( 1, 7,13, 9, 3,11, 5)( 2, 8,14, 6,12, 4,10)$
$ 3, 3, 3, 3, 1, 1 $ $98$ $3$ $( 3,13, 7)( 4, 6,10)( 5, 9,11)( 8,14,12)$
$ 2, 2, 2, 2, 2, 2, 2 $ $21$ $2$ $( 1, 8)( 2, 5)( 3, 4)( 6,13)( 7,14)( 9,12)(10,11)$
$ 14 $ $63$ $14$ $( 1,10,11,12, 9,14, 7, 2, 5, 4, 3, 6,13, 8)$
$ 14 $ $63$ $14$ $( 1,14, 7, 6,13,12, 9, 4, 3,10,11, 2, 5, 8)$
$ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1 $ $14$ $3$ $( 4, 6,10)( 8,14,12)$
$ 7, 3, 3, 1 $ $42$ $21$ $( 1,11, 9, 7, 5, 3,13)( 4, 6,10)( 8,14,12)$
$ 7, 3, 3, 1 $ $42$ $21$ $( 1, 7,13, 9, 3,11, 5)( 4, 6,10)( 8,14,12)$
$ 3, 3, 3, 3, 1, 1 $ $49$ $3$ $( 3,13, 7)( 4,10, 6)( 5, 9,11)( 8,12,14)$
$ 6, 6, 2 $ $147$ $6$ $( 1, 8, 7,14, 9,12)( 2, 5)( 3, 4,13, 6,11,10)$
$ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1 $ $14$ $3$ $( 4,10, 6)( 8,12,14)$
$ 7, 3, 3, 1 $ $42$ $21$ $( 1,11, 9, 7, 5, 3,13)( 4,10, 6)( 8,12,14)$
$ 7, 3, 3, 1 $ $42$ $21$ $( 1, 7,13, 9, 3,11, 5)( 4,10, 6)( 8,12,14)$
$ 3, 3, 3, 3, 1, 1 $ $49$ $3$ $( 3, 7,13)( 4, 6,10)( 5,11, 9)( 8,14,12)$
$ 6, 6, 2 $ $147$ $6$ $( 1, 8, 9,12, 7,14)( 2, 5)( 3, 4,11,10,13, 6)$

Group invariants

Order:  $882=2 \cdot 3^{2} \cdot 7^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [882, 34]
Character table:    
      2  1  .  .  1  .  1  .  1   1   1   .   .   .  1   1   .   .   .  1   1
      3  2  1  1  .  .  .  2  1   .   .   2   1   1  2   1   2   1   1  2   1
      7  2  2  2  2  2  2  .  1   1   1   1   1   1  .   .   1   1   1  .   .

        1a 7a 7b 7c 7d 7e 3a 2a 14a 14b  3b 21a 21b 3c  6a  3d 21c 21d 3e  6b
     2P 1a 7a 7b 7c 7d 7e 3a 1a  7c  7e  3d 21c 21d 3e  3e  3b 21a 21b 3c  3c
     3P 1a 7b 7a 7e 7d 7c 1a 2a 14b 14a  1a  7b  7a 1a  2a  1a  7b  7a 1a  2a
     5P 1a 7b 7a 7e 7d 7c 3a 2a 14b 14a  3d 21d 21c 3e  6b  3b 21b 21a 3c  6a
     7P 1a 1a 1a 1a 1a 1a 3a 2a  2a  2a  3b  3b  3b 3c  6a  3d  3d  3d 3e  6b
    11P 1a 7a 7b 7c 7d 7e 3a 2a 14a 14b  3d 21c 21d 3e  6b  3b 21a 21b 3c  6a
    13P 1a 7b 7a 7e 7d 7c 3a 2a 14b 14a  3b 21b 21a 3c  6a  3d 21d 21c 3e  6b
    17P 1a 7b 7a 7e 7d 7c 3a 2a 14b 14a  3d 21d 21c 3e  6b  3b 21b 21a 3c  6a
    19P 1a 7b 7a 7e 7d 7c 3a 2a 14b 14a  3b 21b 21a 3c  6a  3d 21d 21c 3e  6b

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  1 -1  -1  -1   F   F   F  F  -F  /F  /F  /F /F -/F
X.4      1  1  1  1  1  1  1 -1  -1  -1  /F  /F  /F /F -/F   F   F   F  F  -F
X.5      1  1  1  1  1  1  1  1   1   1   F   F   F  F   F  /F  /F  /F /F  /F
X.6      1  1  1  1  1  1  1  1   1   1  /F  /F  /F /F  /F   F   F   F  F   F
X.7      2  2  2  2  2  2 -1  .   .   .  -1  -1  -1  2   .  -1  -1  -1  2   .
X.8      2  2  2  2  2  2 -1  .   .   .  -F  -F  -F  J   . -/F -/F -/F /J   .
X.9      2  2  2  2  2  2 -1  .   .   . -/F -/F -/F /J   .  -F  -F  -F  J   .
X.10     6  A /A  C -1 /C  .  .   .   .   3 -/E  -E  .   .   3 -/E  -E  .   .
X.11     6 /A  A /C -1  C  .  .   .   .   3  -E -/E  .   .   3  -E -/E  .   .
X.12     6  A /A  C -1 /C  .  .   .   .   G   H  /I  .   .  /G   I  /H  .   .
X.13     6  A /A  C -1 /C  .  .   .   .  /G   I  /H  .   .   G   H  /I  .   .
X.14     6 /A  A /C -1  C  .  .   .   .   G  /I   H  .   .  /G  /H   I  .   .
X.15     6 /A  A /C -1  C  .  .   .   .  /G  /H   I  .   .   G  /I   H  .   .
X.16     9  B /B  D  2 /D  . -3   E  /E   .   .   .  .   .   .   .   .  .   .
X.17     9 /B  B /D  2  D  . -3  /E   E   .   .   .  .   .   .   .   .  .   .
X.18     9  B /B  D  2 /D  .  3  -E -/E   .   .   .  .   .   .   .   .  .   .
X.19     9 /B  B /D  2  D  .  3 -/E  -E   .   .   .  .   .   .   .   .  .   .
X.20    18 -3 -3  4 -3  4  .  .   .   .   .   .   .  .   .   .   .   .  .   .

A = -2*E(7)-2*E(7)^2-3*E(7)^3-2*E(7)^4-3*E(7)^5-3*E(7)^6
  = (5+Sqrt(-7))/2 = 3+b7
B = 3*E(7)^3+3*E(7)^5+3*E(7)^6
  = (-3-3*Sqrt(-7))/2 = -3-3b7
C = 2*E(7)+2*E(7)^2+2*E(7)^4
  = -1+Sqrt(-7) = 2b7
D = 2*E(7)+2*E(7)^2+E(7)^3+2*E(7)^4+E(7)^5+E(7)^6
  = (-3+Sqrt(-7))/2 = -1+b7
E = -E(7)^3-E(7)^5-E(7)^6
  = (1+Sqrt(-7))/2 = 1+b7
F = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3
G = 3*E(3)^2
  = (-3-3*Sqrt(-3))/2 = -3-3b3
H = E(21)^5+E(21)^17+E(21)^20
I = E(21)^10+E(21)^13+E(21)^19
J = 2*E(3)^2
  = -1-Sqrt(-3) = -1-i3