Properties

Label 34T18
Order \(4352\)
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 $t$ :  $18$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,11,21,31,7,18,28,3,13,23,33,9,19,30,5,15,25)(2,12,22,32,8,17,27,4,14,24,34,10,20,29,6,16,26), (1,5,9,13,18,21,25,30,34,4,8,11,16,19,24,27,32)(2,6,10,14,17,22,26,29,33,3,7,12,15,20,23,28,31)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
17:  $C_{17}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 17: $C_{17}$

Low degree siblings

34T18 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $17$ $2$ $( 5, 6)( 9,10)(11,12)(15,16)(21,22)(23,24)(31,32)(33,34)$
$ 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 $ $17$ $2$ $( 5, 6)( 7, 8)(11,12)(19,20)(23,24)(25,26)$
$ 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 $ $17$ $2$ $( 1, 2)( 5, 6)( 9,10)(11,12)(17,18)(23,24)(29,30)(31,32)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $17$ $2$ $( 1, 2)( 5, 6)( 7, 8)(11,12)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(29,30) (33,34)$
$ 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 $ $17$ $2$ $( 9,10)(13,14)(17,18)(21,22)(31,32)(33,34)$
$ 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 $ $17$ $2$ $( 5, 6)(11,12)(13,14)(15,16)(17,18)(23,24)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $17$ $2$ $( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(17,18)(19,20)(21,22)(23,24)(25,26)(31,32) (33,34)$
$ 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 $ $17$ $2$ $( 3, 4)(11,12)(17,18)(21,22)(23,24)(25,26)(27,28)(31,32)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $17$ $2$ $( 3, 4)( 7, 8)( 9,10)(11,12)(15,16)(17,18)(19,20)(23,24)(27,28)(33,34)$
$ 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 $ $17$ $2$ $( 3, 4)( 5, 6)( 7, 8)(17,18)(19,20)(21,22)(27,28)(31,32)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $17$ $2$ $( 1, 2)( 3, 4)(11,12)(15,16)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)$
$ 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 $ $17$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 9,10)(21,22)(25,26)(27,28)(29,30)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $17$ $2$ $( 1, 2)( 3, 4)( 7, 8)( 9,10)(11,12)(19,20)(21,22)(23,24)(27,28)(29,30)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $17$ $2$ $(11,12)(13,14)(15,16)(17,18)(21,22)(23,24)(27,28)(29,30)(31,32)(33,34)$
$ 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 $ $17$ $2$ $( 1, 2)( 3, 4)(11,12)(21,22)(29,30)(31,32)$
$ 17, 17 $ $256$ $17$ $( 1,11,21,31, 7,18,28, 3,13,23,33, 9,19,30, 5,15,25)( 2,12,22,32, 8,17,27, 4, 14,24,34,10,20,29, 6,16,26)$
$ 17, 17 $ $256$ $17$ $( 1,31,28,23,19,15,11, 7, 3,33,30,25,21,18,13, 9, 5)( 2,32,27,24,20,16,12, 8, 4,34,29,26,22,17,14,10, 6)$
$ 17, 17 $ $256$ $17$ $( 1,23,11,33,21, 9,31,19, 7,30,18, 5,28,15, 3,25,13)( 2,24,12,34,22,10,32,20, 8,29,17, 6,27,16, 4,26,14)$
$ 17, 17 $ $256$ $17$ $( 1,33,31,30,28,25,23,21,19,18,15,13,11, 9, 7, 5, 3)( 2,34,32,29,27,26,24,22, 20,17,16,14,12,10, 8, 6, 4)$
$ 17, 17 $ $256$ $17$ $( 1,30,23,18,11, 5,33,28,21,15, 9, 3,31,25,19,13, 7)( 2,29,24,17,12, 6,34,27, 22,16,10, 4,32,26,20,14, 8)$
$ 17, 17 $ $256$ $17$ $( 1,18,33,15,31,13,30,11,28, 9,25, 7,23, 5,21, 3,19)( 2,17,34,16,32,14,29,12, 27,10,26, 8,24, 6,22, 4,20)$
$ 17, 17 $ $256$ $17$ $( 1,15,30, 9,23, 3,18,31,11,25, 5,19,33,13,28, 7,21)( 2,16,29,10,24, 4,17,32, 12,26, 6,20,34,14,27, 8,22)$
$ 17, 17 $ $256$ $17$ $( 1, 9,18,25,33, 7,15,23,31, 5,13,21,30, 3,11,19,28)( 2,10,17,26,34, 8,16,24, 32, 6,14,22,29, 4,12,20,27)$
$ 17, 17 $ $256$ $17$ $( 1,25,15, 5,30,19, 9,33,23,13, 3,28,18, 7,31,21,11)( 2,26,16, 6,29,20,10,34, 24,14, 4,27,17, 8,32,22,12)$
$ 17, 17 $ $256$ $17$ $( 1, 5, 9,13,18,21,25,30,33, 3, 7,11,15,19,23,28,31)( 2, 6,10,14,17,22,26,29, 34, 4, 8,12,16,20,24,27,32)$
$ 17, 17 $ $256$ $17$ $( 1,13,25, 3,15,28, 5,18,30, 7,19,31, 9,21,33,11,23)( 2,14,26, 4,16,27, 6,17, 29, 8,20,32,10,22,34,12,24)$
$ 17, 17 $ $256$ $17$ $( 1, 3, 5, 7, 9,11,13,15,18,19,21,23,25,28,30,31,33)( 2, 4, 6, 8,10,12,14,16, 17,20,22,24,26,27,29,32,34)$
$ 17, 17 $ $256$ $17$ $( 1, 7,13,19,25,31, 3, 9,15,21,28,33, 5,11,18,23,30)( 2, 8,14,20,26,32, 4,10, 16,22,27,34, 6,12,17,24,29)$
$ 17, 17 $ $256$ $17$ $( 1,19, 3,21, 5,23, 7,25, 9,28,11,30,13,31,15,33,18)( 2,20, 4,22, 6,24, 8,26, 10,27,12,29,14,32,16,34,17)$
$ 17, 17 $ $256$ $17$ $( 1,21, 7,28,13,33,19, 5,25,11,31,18, 3,23, 9,30,15)( 2,22, 8,27,14,34,20, 6, 26,12,32,17, 4,24,10,29,16)$
$ 17, 17 $ $256$ $17$ $( 1,28,19,11, 3,30,21,13, 5,31,23,15, 7,33,25,18, 9)( 2,27,20,12, 4,29,22,14, 6,32,24,16, 8,34,26,17,10)$

Group invariants

Order:  $4352=2^{8} \cdot 17$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [4352, 1118911]
Character table: Data not available.