↑

Properties

Label 34T46
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 $t$ :  $46$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,16,28,13)(2,15,27,14)(3,7,26,21,4,8,25,22)(5,33,23,29)(6,34,24,30)(9,18,19,11,10,17,20,12), (1,24)(2,23)(3,22)(4,21)(5,20,6,19)(7,17,8,18)(9,16,10,15)(11,14,12,13)(25,34)(26,33)(27,32,28,31)(29,30)
$|\Aut(F/K)|$:  $2$

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 2, 34T47 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, 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 $ $17$ $2$ $( 3, 4)( 7, 8)( 9,10)(11,12)(17,18)(19,20)(21,22)(25,26)$
$ 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $34$ $2$ $( 1, 2)( 5, 6)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)$
$ 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 $ $34$ $2$ $( 5, 6)( 7, 8)( 9,10)(19,20)(21,22)(23,24)(29,30)(33,34)$
$ 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 $ $34$ $2$ $(13,14)(15,16)(19,20)(27,28)(31,32)(33,34)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $17$ $2$ $( 3, 4)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(21,22)(25,26)(27,28)(31,32) (33,34)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $17$ $2$ $( 3, 4)( 5, 6)(11,12)(13,14)(15,16)(17,18)(19,20)(23,24)(25,26)(27,28)(29,30) (31,32)$
$ 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 $ $34$ $2$ $( 1, 2)( 3, 4)(11,12)(21,22)(29,30)(31,32)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $34$ $2$ $( 1, 2)( 5, 6)( 7, 8)(11,12)(13,14)(15,16)(17,18)(29,30)(31,32)(33,34)$
$ 17, 17 $ $1024$ $17$ $( 1,18,34,15,32,14,29,11,28,10,26, 7,23, 5,21, 3,19)( 2,17,33,16,31,13,30,12, 27, 9,25, 8,24, 6,22, 4,20)$
$ 17, 17 $ $1024$ $17$ $( 1,15,29,10,23, 3,18,32,11,26, 5,19,34,14,28, 7,21)( 2,16,30, 9,24, 4,17,31, 12,25, 6,20,33,13,27, 8,22)$
$ 17, 17 $ $1024$ $17$ $( 1,10,18,26,34, 7,15,23,32, 5,14,21,29, 3,11,19,28)( 2, 9,17,25,33, 8,16,24, 31, 6,13,22,30, 4,12,20,27)$
$ 17, 17 $ $1024$ $17$ $( 1,26,15, 5,29,19,10,34,23,14, 3,28,18, 7,32,21,11)( 2,25,16, 6,30,20, 9,33, 24,13, 4,27,17, 8,31,22,12)$
$ 4, 4, 4, 4, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $544$ $4$ $( 1, 5)( 2, 6)( 7,34, 8,33)( 9,32,10,31)(11,30)(12,29)(13,27)(14,28) (15,26,16,25)(17,24)(18,23)(19,21,20,22)$
$ 4, 4, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $544$ $4$ $( 1, 5)( 2, 6)( 3, 4)( 7,34)( 8,33)( 9,32)(10,31)(11,30,12,29)(13,27)(14,28) (15,25)(16,26)(17,24,18,23)(19,22,20,21)$
$ 4, 4, 4, 4, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $544$ $4$ $( 1, 6, 2, 5)( 7,33)( 8,34)( 9,32)(10,31)(11,30,12,29)(13,27,14,28) (15,25,16,26)(17,24)(18,23)(19,21)(20,22)$
$ 4, 4, 4, 4, 4, 2, 2, 2, 2, 2, 2, 2 $ $544$ $4$ $( 1, 6, 2, 5)( 3, 4)( 7,33, 8,34)( 9,32,10,31)(11,30)(12,29)(13,27,14,28) (15,26)(16,25)(17,24,18,23)(19,22)(20,21)$
$ 4, 4, 4, 4, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $272$ $4$ $( 1, 5)( 2, 6)( 7,34)( 8,33)( 9,32,10,31)(11,29)(12,30)(13,27,14,28) (15,25,16,26)(17,23,18,24)(19,22)(20,21)$
$ 4, 4, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $544$ $4$ $( 1, 5)( 2, 6)( 3, 4)( 7,34, 8,33)( 9,32)(10,31)(11,29,12,30)(13,27,14,28) (15,26)(16,25)(17,23)(18,24)(19,21)(20,22)$
$ 4, 4, 4, 4, 4, 4, 2, 2, 2, 2, 1, 1 $ $272$ $4$ $( 1, 6, 2, 5)( 7,33, 8,34)( 9,32)(10,31)(11,29,12,30)(13,27)(14,28) (15,26,16,25)(17,23,18,24)(19,22,20,21)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $272$ $2$ $( 1, 5)( 2, 6)( 7,33)( 8,34)( 9,31)(10,32)(11,30)(12,29)(13,28)(14,27)(15,26) (16,25)(17,24)(18,23)(19,21)(20,22)$
$ 4, 4, 4, 4, 4, 2, 2, 2, 2, 2, 2, 2 $ $544$ $4$ $( 1, 5)( 2, 6)( 3, 4)( 7,33, 8,34)( 9,31,10,32)(11,30,12,29)(13,28)(14,27) (15,25,16,26)(17,24,18,23)(19,22)(20,21)$
$ 4, 4, 4, 4, 4, 4, 2, 2, 2, 2, 1, 1 $ $272$ $4$ $( 1, 6, 2, 5)( 7,34, 8,33)( 9,31,10,32)(11,30,12,29)(13,28,14,27)(15,25) (16,26)(17,24)(18,23)(19,21,20,22)$
$ 8, 8, 4, 4, 4, 4, 1, 1 $ $1088$ $8$ $( 1,12, 6,29)( 2,11, 5,30)( 7,21,33,19)( 8,22,34,20)( 9,14,31,27,10,13,32,28) (15,24,25,17,16,23,26,18)$
$ 8, 8, 8, 4, 4, 2 $ $1088$ $8$ $( 1,11, 5,30, 2,12, 6,29)( 3, 4)( 7,22,34,19, 8,21,33,20)( 9,14,31,27) (10,13,32,28)(15,24,26,17,16,23,25,18)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 1, 1 $ $1088$ $4$ $( 1,11, 5,29)( 2,12, 6,30)( 7,22,34,20)( 8,21,33,19)( 9,13,32,28)(10,14,31,27) (15,23,25,17)(16,24,26,18)$
$ 8, 8, 8, 4, 4, 2 $ $1088$ $8$ $( 1,12, 6,30, 2,11, 5,29)( 3, 4)( 7,21,33,20, 8,22,34,19)( 9,13,32,28,10,14, 31,27)(15,23,26,17)(16,24,25,18)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 1, 1 $ $1088$ $4$ $( 1,29, 5,11)( 2,30, 6,12)( 7,20,33,21)( 8,19,34,22)( 9,28,32,14)(10,27,31,13) (15,17,25,24)(16,18,26,23)$
$ 8, 8, 8, 4, 4, 2 $ $1088$ $8$ $( 1,29, 5,12, 2,30, 6,11)( 3, 4)( 7,19,34,21, 8,20,33,22)( 9,28,32,14,10,27, 31,13)(15,18,25,24)(16,17,26,23)$
$ 8, 8, 4, 4, 4, 4, 1, 1 $ $1088$ $8$ $( 1,30, 6,11)( 2,29, 5,12)( 7,20,33,22)( 8,19,34,21)( 9,28,32,13,10,27,31,14) (15,17,26,24,16,18,25,23)$
$ 8, 8, 8, 4, 4, 2 $ $1088$ $8$ $( 1,30, 6,12, 2,29, 5,11)( 3, 4)( 7,19,34,22, 8,20,33,21)( 9,28,32,13) (10,27,31,14)(15,18,26,24,16,17,25,23)$

Group invariants

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