Properties

Label 34T30
Order \(8704\)
n \(34\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No

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Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
34:  $D_{17}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 17: $D_{17}$

Low degree siblings

34T30 x 14, 34T31 x 15

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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $17$ $2$ $( 7, 8)( 9,10)(11,12)(13,14)(21,22)(33,34)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $17$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 9,10)(11,12)(13,14)(15,16)(21,22)(27,28)(33,34)$
$ 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 $ $17$ $2$ $( 1, 2)( 3, 4)( 9,10)(11,12)(17,18)(19,20)(21,22)(33,34)$
$ 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 $ $17$ $2$ $( 1, 2)( 3, 4)( 7, 8)(13,14)(17,18)(19,20)$
$ 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 $ $17$ $2$ $( 3, 4)(11,12)(17,18)(19,20)(29,30)(31,32)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $17$ $2$ $( 3, 4)( 7, 8)( 9,10)(13,14)(17,18)(19,20)(21,22)(29,30)(31,32)(33,34)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $17$ $2$ $( 1, 2)( 5, 6)( 9,10)(13,14)(15,16)(17,18)(19,20)(21,22)(27,28)(29,30)(31,32) (33,34)$
$ 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 $ $17$ $2$ $( 3, 4)(17,18)(21,22)(23,24)(25,26)(29,30)(31,32)(33,34)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $17$ $2$ $( 3, 4)( 7, 8)( 9,10)(11,12)(13,14)(17,18)(23,24)(25,26)(29,30)(31,32)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $17$ $2$ $( 1, 2)( 5, 6)( 7, 8)(15,16)(17,18)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32) (33,34)$
$ 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 $ $17$ $2$ $( 1, 2)( 9,10)(11,12)(19,20)(23,24)(25,26)(29,30)(31,32)$
$ 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 $ $17$ $2$ $( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(27,28)(31,32)$
$ 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 $ $17$ $2$ $( 1, 2)( 3, 4)( 9,10)(11,12)(15,16)(19,20)(27,28)(31,32)$
$ 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 $ $17$ $2$ $( 1, 2)( 5, 6)( 9,10)(13,14)(19,20)(29,30)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $17$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 9,10)(11,12)(15,16)(17,18)(19,20)(25,26)(29,30)$
$ 17, 17 $ $512$ $17$ $( 1,25,16, 6,30,19,10,33,23,14, 3,27,17, 8,32,21,11)( 2,26,15, 5,29,20, 9,34, 24,13, 4,28,18, 7,31,22,12)$
$ 17, 17 $ $512$ $17$ $( 1, 6,10,14,17,21,25,30,33, 3, 8,11,16,19,23,27,32)( 2, 5, 9,13,18,22,26,29, 34, 4, 7,12,15,20,24,28,31)$
$ 17, 17 $ $512$ $17$ $( 1,14,25, 3,16,27, 6,17,30, 8,19,32,10,21,33,11,23)( 2,13,26, 4,15,28, 5,18, 29, 7,20,31, 9,22,34,12,24)$
$ 17, 17 $ $512$ $17$ $( 1, 3, 6, 8,10,11,14,16,17,19,21,23,25,27,30,32,33)( 2, 4, 5, 7, 9,12,13,15, 18,20,22,24,26,28,29,31,34)$
$ 17, 17 $ $512$ $17$ $( 1, 8,14,19,25,32, 3,10,16,21,27,33, 6,11,17,23,30)( 2, 7,13,20,26,31, 4, 9, 15,22,28,34, 5,12,18,24,29)$
$ 17, 17 $ $512$ $17$ $( 1,19, 3,21, 6,23, 8,25,10,27,11,30,14,32,16,33,17)( 2,20, 4,22, 5,24, 7,26, 9,28,12,29,13,31,15,34,18)$
$ 17, 17 $ $512$ $17$ $( 1,21, 8,27,14,33,19, 6,25,11,32,17, 3,23,10,30,16)( 2,22, 7,28,13,34,20, 5, 26,12,31,18, 4,24, 9,29,15)$
$ 17, 17 $ $512$ $17$ $( 1,27,19,11, 3,30,21,14, 6,32,23,16, 8,33,25,17,10)( 2,28,20,12, 4,29,22,13, 5,31,24,15, 7,34,26,18, 9)$
$ 4, 4, 4, 4, 4, 4, 2, 2, 2, 2, 1, 1 $ $272$ $4$ $( 1,14, 2,13)( 3,12, 4,11)( 5,10)( 6, 9)(15,33)(16,34)(17,31,18,32) (19,30,20,29)(21,27,22,28)(23,25,24,26)$
$ 4, 4, 4, 4, 4, 2, 2, 2, 2, 2, 2, 2 $ $272$ $4$ $( 1,13)( 2,14)( 3,11)( 4,12)( 5, 9, 6,10)( 7, 8)(15,34,16,33)(17,31,18,32) (19,30,20,29)(21,27)(22,28)(23,25,24,26)$
$ 4, 4, 4, 4, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $272$ $4$ $( 1,13)( 2,14)( 3,12, 4,11)( 5,10, 6, 9)(15,33,16,34)(17,31)(18,32)(19,30) (20,29)(21,28)(22,27)(23,25,24,26)$
$ 4, 4, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $272$ $4$ $( 1,14, 2,13)( 3,11)( 4,12)( 5, 9)( 6,10)( 7, 8)(15,34)(16,33)(17,31)(18,32) (19,30)(20,29)(21,28,22,27)(23,25,24,26)$
$ 4, 4, 4, 4, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $272$ $4$ $( 1,14, 2,13)( 3,11)( 4,12)( 5,10)( 6, 9)(15,34,16,33)(17,31,18,32)(19,30) (20,29)(21,27)(22,28)(23,26,24,25)$
$ 4, 4, 4, 4, 4, 2, 2, 2, 2, 2, 2, 2 $ $272$ $4$ $( 1,13)( 2,14)( 3,12, 4,11)( 5, 9, 6,10)( 7, 8)(15,33)(16,34)(17,31,18,32) (19,30)(20,29)(21,27,22,28)(23,26,24,25)$
$ 4, 4, 4, 4, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $272$ $4$ $( 1,13)( 2,14)( 3,11)( 4,12)( 5,10, 6, 9)(15,34)(16,33)(17,31)(18,32) (19,30,20,29)(21,28,22,27)(23,26,24,25)$
$ 4, 4, 4, 4, 4, 2, 2, 2, 2, 2, 2, 2 $ $272$ $4$ $( 1,14, 2,13)( 3,12, 4,11)( 5, 9)( 6,10)( 7, 8)(15,33,16,34)(17,31)(18,32) (19,30,20,29)(21,28)(22,27)(23,26,24,25)$
$ 4, 4, 4, 4, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $272$ $4$ $( 1,13)( 2,14)( 3,12, 4,11)( 5,10)( 6, 9)(15,34,16,33)(17,31)(18,32) (19,29,20,30)(21,28,22,27)(23,26)(24,25)$
$ 4, 4, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $272$ $4$ $( 1,14, 2,13)( 3,11)( 4,12)( 5, 9, 6,10)( 7, 8)(15,33)(16,34)(17,31)(18,32) (19,29,20,30)(21,28)(22,27)(23,26)(24,25)$
$ 4, 4, 4, 4, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $272$ $4$ $( 1,14, 2,13)( 3,12, 4,11)( 5,10, 6, 9)(15,34)(16,33)(17,31,18,32)(19,29) (20,30)(21,27)(22,28)(23,26)(24,25)$
$ 4, 4, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $272$ $4$ $( 1,13)( 2,14)( 3,11)( 4,12)( 5, 9)( 6,10)( 7, 8)(15,33,16,34)(17,31,18,32) (19,29)(20,30)(21,27,22,28)(23,26)(24,25)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $272$ $2$ $( 1,13)( 2,14)( 3,11)( 4,12)( 5,10)( 6, 9)(15,33)(16,34)(17,31)(18,32)(19,29) (20,30)(21,28)(22,27)(23,25)(24,26)$
$ 4, 4, 4, 4, 4, 2, 2, 2, 2, 2, 2, 2 $ $272$ $4$ $( 1,14, 2,13)( 3,12, 4,11)( 5, 9, 6,10)( 7, 8)(15,34,16,33)(17,31)(18,32) (19,29)(20,30)(21,28,22,27)(23,25)(24,26)$
$ 4, 4, 4, 4, 4, 4, 2, 2, 2, 2, 1, 1 $ $272$ $4$ $( 1,14, 2,13)( 3,11)( 4,12)( 5,10, 6, 9)(15,33,16,34)(17,31,18,32) (19,29,20,30)(21,27,22,28)(23,25)(24,26)$
$ 4, 4, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $272$ $4$ $( 1,13)( 2,14)( 3,12, 4,11)( 5, 9)( 6,10)( 7, 8)(15,34)(16,33)(17,31,18,32) (19,29,20,30)(21,27)(22,28)(23,25)(24,26)$

Group invariants

Order:  $8704=2^{9} \cdot 17$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  Data not available
Character table: Data not available.