Properties

Label 28T35
Order \(196\)
n \(28\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_7:D_7.C_2$

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Group action invariants

Degree $n$ :  $28$
Transitive number $t$ :  $35$
Group :  $C_7:D_7.C_2$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,9)(2,10)(3,11)(4,12)(5,6)(7,8)(13,25)(14,26)(15,27)(16,28)(17,22)(18,21)(19,24)(20,23), (1,28,21,8)(2,27,22,7)(3,25,4,26)(5,24,18,11)(6,23,17,12)(9,20,14,15)(10,19,13,16)
$|\Aut(F/K)|$:  $14$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
4:  $C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$

Degree 7: None

Degree 14: 14T12

Low degree siblings

14T12 x 4, 28T35 x 3

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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 7, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $4$ $7$ $( 3, 8,12,16,20,24,27)( 4, 7,11,15,19,23,28)$
$ 7, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $4$ $7$ $( 3,12,20,27, 8,16,24)( 4,11,19,28, 7,15,23)$
$ 7, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $4$ $7$ $( 3,16,27,12,24, 8,20)( 4,15,28,11,23, 7,19)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $49$ $2$ $( 1, 2)( 3, 4)( 5,25)( 6,26)( 7,27)( 8,28)( 9,22)(10,21)(11,24)(12,23)(13,18) (14,17)(15,20)(16,19)$
$ 4, 4, 4, 4, 4, 4, 4 $ $49$ $4$ $( 1, 3, 2, 4)( 5, 7,25,27)( 6, 8,26,28)( 9,11,22,24)(10,12,21,23)(13,15,18,20) (14,16,17,19)$
$ 4, 4, 4, 4, 4, 4, 4 $ $49$ $4$ $( 1, 4, 2, 3)( 5,27,25, 7)( 6,28,26, 8)( 9,24,22,11)(10,23,21,12)(13,20,18,15) (14,19,17,16)$
$ 7, 7, 7, 7 $ $4$ $7$ $( 1, 6,10,14,18,22,25)( 2, 5, 9,13,17,21,26)( 3, 8,12,16,20,24,27) ( 4, 7,11,15,19,23,28)$
$ 7, 7, 7, 7 $ $4$ $7$ $( 1, 6,10,14,18,22,25)( 2, 5, 9,13,17,21,26)( 3,12,20,27, 8,16,24) ( 4,11,19,28, 7,15,23)$
$ 7, 7, 7, 7 $ $4$ $7$ $( 1, 6,10,14,18,22,25)( 2, 5, 9,13,17,21,26)( 3,16,27,12,24, 8,20) ( 4,15,28,11,23, 7,19)$
$ 7, 7, 7, 7 $ $4$ $7$ $( 1, 6,10,14,18,22,25)( 2, 5, 9,13,17,21,26)( 3,20, 8,24,12,27,16) ( 4,19, 7,23,11,28,15)$
$ 7, 7, 7, 7 $ $4$ $7$ $( 1, 6,10,14,18,22,25)( 2, 5, 9,13,17,21,26)( 3,24,16, 8,27,20,12) ( 4,23,15, 7,28,19,11)$
$ 7, 7, 7, 7 $ $4$ $7$ $( 1,10,18,25, 6,14,22)( 2, 9,17,26, 5,13,21)( 3,12,20,27, 8,16,24) ( 4,11,19,28, 7,15,23)$
$ 7, 7, 7, 7 $ $4$ $7$ $( 1,10,18,25, 6,14,22)( 2, 9,17,26, 5,13,21)( 3,16,27,12,24, 8,20) ( 4,15,28,11,23, 7,19)$
$ 7, 7, 7, 7 $ $4$ $7$ $( 1,10,18,25, 6,14,22)( 2, 9,17,26, 5,13,21)( 3,20, 8,24,12,27,16) ( 4,19, 7,23,11,28,15)$
$ 7, 7, 7, 7 $ $4$ $7$ $( 1,14,25,10,22, 6,18)( 2,13,26, 9,21, 5,17)( 3,16,27,12,24, 8,20) ( 4,15,28,11,23, 7,19)$

Group invariants

Order:  $196=2^{2} \cdot 7^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [196, 8]
Character table:   
      2  2  .  .  .  2  2  2  .  .  .  .  .  .  .  .  .
      7  2  2  2  2  .  .  .  2  2  2  2  2  2  2  2  2

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

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

A = -2*E(7)-E(7)^2-2*E(7)^3-2*E(7)^4-E(7)^5-2*E(7)^6
B = -E(7)-2*E(7)^2-2*E(7)^3-2*E(7)^4-2*E(7)^5-E(7)^6
C = -2*E(7)-2*E(7)^2-E(7)^3-E(7)^4-2*E(7)^5-2*E(7)^6
D = 2*E(7)^2+2*E(7)^5
E = 2*E(7)^3+2*E(7)^4
F = 2*E(7)+2*E(7)^6
G = E(7)^2+E(7)^3+E(7)^4+E(7)^5
H = E(7)+E(7)^3+E(7)^4+E(7)^6
I = E(7)+E(7)^2+E(7)^5+E(7)^6
J = -E(4)
  = -Sqrt(-1) = -i