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Properties

Label 34T6
Order \(136\)
n \(34\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_{17}:C_8$

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Group action invariants

Degree $n$ :  $34$
Transitive number $t$ :  $6$
Group :  $C_{17}:C_8$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,28,24,21,4,12,15,17)(2,27,23,22,3,11,16,18)(5,30,7,14,34,9,31,25)(6,29,8,13,33,10,32,26)(19,20), (1,20)(2,19)(3,17)(4,18)(5,16)(6,15)(7,13)(8,14)(9,11)(10,12)(21,34)(22,33)(23,31)(24,32)(25,30)(26,29)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
4:  $C_4$
8:  $C_8$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 17: $C_{17}:C_{8}$

Low degree siblings

17T4

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, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 4, 4, 4, 4, 4, 4, 4, 4, 1, 1 $ $17$ $4$ $( 3,10,34,28)( 4, 9,33,27)( 5,17,31,19)( 6,18,32,20)( 7,26,29,12)( 8,25,30,11) (13,16,23,21)(14,15,24,22)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 1, 1 $ $17$ $4$ $( 3,28,34,10)( 4,27,33, 9)( 5,19,31,17)( 6,20,32,18)( 7,12,29,26)( 8,11,30,25) (13,21,23,16)(14,22,24,15)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $17$ $2$ $( 3,34)( 4,33)( 5,31)( 6,32)( 7,29)( 8,30)( 9,27)(10,28)(11,25)(12,26)(13,23) (14,24)(15,22)(16,21)(17,19)(18,20)$
$ 8, 8, 8, 8, 2 $ $17$ $8$ $( 1, 2)( 3, 6,10,18,34,32,28,20)( 4, 5, 9,17,33,31,27,19)( 7,14,26,15,29,24, 12,22)( 8,13,25,16,30,23,11,21)$
$ 8, 8, 8, 8, 2 $ $17$ $8$ $( 1, 2)( 3,18,28, 6,34,20,10,32)( 4,17,27, 5,33,19, 9,31)( 7,15,12,14,29,22, 26,24)( 8,16,11,13,30,21,25,23)$
$ 8, 8, 8, 8, 2 $ $17$ $8$ $( 1, 2)( 3,20,28,32,34,18,10, 6)( 4,19,27,31,33,17, 9, 5)( 7,22,12,24,29,15, 26,14)( 8,21,11,23,30,16,25,13)$
$ 8, 8, 8, 8, 2 $ $17$ $8$ $( 1, 2)( 3,32,10,20,34, 6,28,18)( 4,31, 9,19,33, 5,27,17)( 7,24,26,22,29,14, 12,15)( 8,23,25,21,30,13,11,16)$
$ 17, 17 $ $8$ $17$ $( 1, 4, 6, 8, 9,11,14,15,18,20,22,24,25,27,30,32,33)( 2, 3, 5, 7,10,12,13,16, 17,19,21,23,26,28,29,31,34)$
$ 17, 17 $ $8$ $17$ $( 1, 8,14,20,25,32, 4, 9,15,22,27,33, 6,11,18,24,30)( 2, 7,13,19,26,31, 3,10, 16,21,28,34, 5,12,17,23,29)$

Group invariants

Order:  $136=2^{3} \cdot 17$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [136, 12]
Character table:    
      2  3  3  3  3   3   3   3   3   .   .
     17  1  .  .  .   .   .   .   .   1   1

        1a 4a 4b 2a  8a  8b  8c  8d 17a 17b
     2P 1a 2a 2a 1a  4a  4b  4b  4a 17a 17b
     3P 1a 4b 4a 2a  8b  8a  8d  8c 17b 17a
     5P 1a 4a 4b 2a  8d  8c  8b  8a 17b 17a
     7P 1a 4b 4a 2a  8c  8d  8a  8b 17b 17a
    11P 1a 4b 4a 2a  8b  8a  8d  8c 17b 17a
    13P 1a 4a 4b 2a  8d  8c  8b  8a 17a 17b
    17P 1a 4a 4b 2a  8a  8b  8c  8d  1a  1a

X.1      1  1  1  1   1   1   1   1   1   1
X.2      1  1  1  1  -1  -1  -1  -1   1   1
X.3      1 -1 -1  1   A  -A  -A   A   1   1
X.4      1 -1 -1  1  -A   A   A  -A   1   1
X.5      1  A -A -1   B -/B  /B  -B   1   1
X.6      1  A -A -1  -B  /B -/B   B   1   1
X.7      1 -A  A -1 -/B   B  -B  /B   1   1
X.8      1 -A  A -1  /B  -B   B -/B   1   1
X.9      8  .  .  .   .   .   .   .   C  *C
X.10     8  .  .  .   .   .   .   .  *C   C

A = -E(4)
  = -Sqrt(-1) = -i
B = -E(8)^3
C = E(17)^3+E(17)^5+E(17)^6+E(17)^7+E(17)^10+E(17)^11+E(17)^12+E(17)^14
  = (-1-Sqrt(17))/2 = -1-b17