Properties

Label 34T31
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 $t$ :  $31$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,28)(2,27)(3,25)(4,26)(5,23)(6,24)(7,22)(8,21)(9,20)(10,19)(11,17)(12,18)(13,15)(14,16)(29,34)(30,33)(31,32), (1,30,24,18,12,6,34,28,22,15,9,4,32,25,20,14,8)(2,29,23,17,11,5,33,27,21,16,10,3,31,26,19,13,7)
$|\Aut(F/K)|$:  $2$

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 15, 34T31 x 14

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

Group invariants

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