↑

Properties

Label 34T47
Order \(17408\)
n \(34\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No

Learn more about

Group action invariants

Degree $n$ :  $34$
Transitive number $t$ :  $47$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,26,32,8)(2,25,31,7)(3,17,30,15)(4,18,29,16)(5,9,27,23,6,10,28,24)(11,19,22,14,12,20,21,13)(33,34), (1,33,25,28,2,34,26,27)(3,8,23,20)(4,7,24,19)(5,16,21,12,6,15,22,11)(9,32,17,29,10,31,18,30)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
4:  $C_4$
68:  $C_{17}:C_{4}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 17: $C_{17}:C_{4}$

Low degree siblings

34T46 x 3, 34T47 x 2

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 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $17$ $2$ $( 5, 6)( 9,10)(11,12)(13,14)(19,20)(21,22)(23,24)(27,28)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $34$ $2$ $( 1, 2)( 3, 4)( 9,10)(15,16)(21,22)(23,24)(27,28)(31,32)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $34$ $2$ $( 1, 2)( 7, 8)(11,12)(15,16)(17,18)(19,20)(23,24)(25,26)(27,28)(31,32)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $34$ $2$ $(15,16)(21,22)(23,24)(25,26)(27,28)(33,34)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $34$ $2$ $( 5, 6)( 9,10)(11,12)(13,14)(15,16)(19,20)(25,26)(33,34)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $17$ $2$ $( 1, 2)( 3, 4)( 5, 6)(11,12)(13,14)(19,20)(21,22)(23,24)(25,26)(27,28)(31,32) (33,34)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $34$ $2$ $( 7, 8)(11,12)(13,14)(27,28)(29,30)(33,34)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $34$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)(15,16)(19,20)(27,28)(29,30)(31,32)(33,34)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $17$ $2$ $( 1, 2)( 5, 6)( 9,10)(11,12)(15,16)(17,18)(21,22)(25,26)(27,28)(29,30)(31,32) (33,34)$
$ 17, 17 $ $1024$ $17$ $( 1,34,32,29,28,25,24,21,19,18,16,14,12,10, 7, 6, 3)( 2,33,31,30,27,26,23,22, 20,17,15,13,11, 9, 8, 5, 4)$
$ 17, 17 $ $1024$ $17$ $( 1,29,24,18,12, 6,34,28,21,16,10, 3,32,25,19,14, 7)( 2,30,23,17,11, 5,33,27, 22,15, 9, 4,31,26,20,13, 8)$
$ 17, 17 $ $1024$ $17$ $( 1,18,34,16,32,14,29,12,28,10,25, 7,24, 6,21, 3,19)( 2,17,33,15,31,13,30,11, 27, 9,26, 8,23, 5,22, 4,20)$
$ 17, 17 $ $1024$ $17$ $( 1,16,29,10,24, 3,18,32,12,25, 6,19,34,14,28, 7,21)( 2,15,30, 9,23, 4,17,31, 11,26, 5,20,33,13,27, 8,22)$
$ 4, 4, 4, 4, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $544$ $4$ $( 1,10)( 2, 9)( 3, 8, 4, 7)(11,33)(12,34)(13,32,14,31)(15,30,16,29) (17,27,18,28)(19,25)(20,26)(21,23)(22,24)$
$ 4, 4, 4, 4, 4, 2, 2, 2, 2, 2, 2, 2 $ $544$ $4$ $( 1, 9, 2,10)( 3, 8, 4, 7)( 5, 6)(11,33,12,34)(13,32)(14,31)(15,30,16,29) (17,28)(18,27)(19,25,20,26)(21,24)(22,23)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $272$ $2$ $( 1, 9)( 2,10)( 3, 8)( 4, 7)(11,33)(12,34)(13,31)(14,32)(15,30)(16,29)(17,28) (18,27)(19,25)(20,26)(21,24)(22,23)$
$ 4, 4, 4, 4, 4, 2, 2, 2, 2, 2, 2, 2 $ $544$ $4$ $( 1,10, 2, 9)( 3, 8)( 4, 7)( 5, 6)(11,33,12,34)(13,31,14,32)(15,30)(16,29) (17,27,18,28)(19,25,20,26)(21,23)(22,24)$
$ 4, 4, 4, 4, 4, 4, 2, 2, 2, 2, 1, 1 $ $272$ $4$ $( 1, 9, 2,10)( 3, 7, 4, 8)(11,33,12,34)(13,32,14,31)(15,30,16,29)(17,27) (18,28)(19,26)(20,25)(21,23,22,24)$
$ 4, 4, 4, 4, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $544$ $4$ $( 1,10, 2, 9)( 3, 7)( 4, 8)(11,33,12,34)(13,31)(14,32)(15,30)(16,29) (17,28,18,27)(19,26)(20,25)(21,24,22,23)$
$ 4, 4, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $544$ $4$ $( 1, 9)( 2,10)( 3, 7)( 4, 8)( 5, 6)(11,33)(12,34)(13,31,14,32)(15,30)(16,29) (17,27)(18,28)(19,26,20,25)(21,23,22,24)$
$ 4, 4, 4, 4, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $272$ $4$ $( 1, 9)( 2,10)( 3, 8)( 4, 7)(11,34,12,33)(13,31)(14,32)(15,30,16,29) (17,27,18,28)(19,26,20,25)(21,23)(22,24)$
$ 4, 4, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $544$ $4$ $( 1,10, 2, 9)( 3, 8)( 4, 7)( 5, 6)(11,34)(12,33)(13,31,14,32)(15,30,16,29) (17,28)(18,27)(19,26)(20,25)(21,24)(22,23)$
$ 4, 4, 4, 4, 4, 4, 2, 2, 2, 2, 1, 1 $ $272$ $4$ $( 1, 9, 2,10)( 3, 7, 4, 8)(11,34)(12,33)(13,32,14,31)(15,30)(16,29) (17,28,18,27)(19,25,20,26)(21,24,22,23)$
$ 8, 8, 8, 4, 4, 1, 1 $ $1088$ $8$ $( 1,22,10,24, 2,21, 9,23)( 3,13, 8,32, 4,14, 7,31)(11,16,33,29,12,15,34,30) (17,26,28,19)(18,25,27,20)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 2 $ $1088$ $4$ $( 1,21,10,23)( 2,22, 9,24)( 3,14, 7,31)( 4,13, 8,32)( 5, 6)(11,16,33,29) (12,15,34,30)(17,26,27,19)(18,25,28,20)$
$ 8, 8, 8, 4, 4, 1, 1 $ $1088$ $8$ $( 1,21,10,23, 2,22, 9,24)( 3,13, 8,31, 4,14, 7,32)(11,15,34,30)(12,16,33,29) (17,26,27,20,18,25,28,19)$
$ 8, 8, 4, 4, 4, 4, 2 $ $1088$ $8$ $( 1,22,10,24)( 2,21, 9,23)( 3,14, 7,32)( 4,13, 8,31)( 5, 6)(11,15,34,30,12,16, 33,29)(17,26,28,20,18,25,27,19)$
$ 8, 8, 8, 4, 4, 1, 1 $ $1088$ $8$ $( 1,24, 9,21, 2,23,10,22)( 3,32, 7,13, 4,31, 8,14)(11,29,34,16)(12,30,33,15) (17,20,28,25,18,19,27,26)$
$ 8, 8, 4, 4, 4, 4, 2 $ $1088$ $8$ $( 1,23, 9,22)( 2,24,10,21)( 3,32, 7,14)( 4,31, 8,13)( 5, 6)(11,29,34,16,12,30, 33,15)(17,19,28,25,18,20,27,26)$
$ 8, 8, 8, 4, 4, 1, 1 $ $1088$ $8$ $( 1,23, 9,22, 2,24,10,21)( 3,31, 8,14, 4,32, 7,13)(11,29,34,15,12,30,33,16) (17,20,27,26)(18,19,28,25)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 2 $ $1088$ $4$ $( 1,24, 9,21)( 2,23,10,22)( 3,31, 8,13)( 4,32, 7,14)( 5, 6)(11,29,34,15) (12,30,33,16)(17,19,27,26)(18,20,28,25)$

Group invariants

Order:  $17408=2^{10} \cdot 17$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  Data not available
Character table: Data not available.