Properties

Label 34T35
Order \(9248\)
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 $t$ :  $35$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,14,16,15,7,11,9,10)(2,5,12,17,6,3,13,8)(18,21,28,33,22,19,29,24)(23,27,25,26,34,30,32,31), (1,5,14,13,15,11,2,3)(4,16,9,6,12,17,7,10)(18,22,30,29,27,23,32,33)(19,24,34,20,26,21,28,25), (1,25,17,19,13,29,14,18)(2,31,4,26,12,23,10,28)(3,20,8,33,11,34,6,21)(5,32,16,30,9,22,15,24)(7,27)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 7
4:  $C_4$ x 4, $C_2^2$ x 7
8:  $C_4\times C_2$ x 6, $C_2^3$
16:  $C_4\times C_2^2$
32:  $(C_8:C_2):C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 17: None

Low degree siblings

34T35 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 $ $1$ $1$ $()$
$ 17, 17 $ $32$ $17$ $( 1, 4, 7,10,13,16, 2, 5, 8,11,14,17, 3, 6, 9,12,15)(18,28,21,31,24,34,27,20, 30,23,33,26,19,29,22,32,25)$
$ 17, 17 $ $32$ $17$ $( 1,10, 2,11, 3,12, 4,13, 5,14, 6,15, 7,16, 8,17, 9)(18,31,27,23,19,32,28,24, 20,33,29,25,21,34,30,26,22)$
$ 17, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $16$ $17$ $(18,27,19,28,20,29,21,30,22,31,23,32,24,33,25,34,26)$
$ 17, 17 $ $16$ $17$ $( 1, 4, 7,10,13,16, 2, 5, 8,11,14,17, 3, 6, 9,12,15)(18,20,22,24,26,28,30,32, 34,19,21,23,25,27,29,31,33)$
$ 17, 17 $ $32$ $17$ $( 1,10, 2,11, 3,12, 4,13, 5,14, 6,15, 7,16, 8,17, 9)(18,23,28,33,21,26,31,19, 24,29,34,22,27,32,20,25,30)$
$ 17, 17 $ $32$ $17$ $( 1,11, 4,14, 7,17,10, 3,13, 6,16, 9, 2,12, 5,15, 8)(18,32,29,26,23,20,34,31, 28,25,22,19,33,30,27,24,21)$
$ 17, 17 $ $32$ $17$ $( 1, 6,11,16, 4, 9,14, 2, 7,12,17, 5,10,15, 3, 8,13)(18,21,24,27,30,33,19,22, 25,28,31,34,20,23,26,29,32)$
$ 17, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $16$ $17$ $( 1,17,16,15,14,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)$
$ 17, 17 $ $32$ $17$ $( 1,15,12, 9, 6, 3,17,14,11, 8, 5, 2,16,13,10, 7, 4)(18,34,33,32,31,30,29,28, 27,26,25,24,23,22,21,20,19)$
$ 17, 17 $ $16$ $17$ $( 1, 8,15, 5,12, 2, 9,16, 6,13, 3,10,17, 7,14, 4,11)(18,22,26,30,34,21,25,29, 33,20,24,28,32,19,23,27,31)$
$ 17, 17 $ $16$ $17$ $( 1, 3, 5, 7, 9,11,13,15,17, 2, 4, 6, 8,10,12,14,16)(18,28,21,31,24,34,27,20, 30,23,33,26,19,29,22,32,25)$
$ 17, 17 $ $16$ $17$ $( 1, 5, 9,13,17, 4, 8,12,16, 3, 7,11,15, 2, 6,10,14)(18,30,25,20,32,27,22,34, 29,24,19,31,26,21,33,28,23)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $289$ $2$ $( 2,17)( 3,16)( 4,15)( 5,14)( 6,13)( 7,12)( 8,11)( 9,10)(19,34)(20,33)(21,32) (22,31)(23,30)(24,29)(25,28)(26,27)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 1, 1 $ $289$ $4$ $( 2,14,17, 5)( 3,10,16, 9)( 4, 6,15,13)( 7,11,12, 8)(19,31,34,22)(20,27,33,26) (21,23,32,30)(24,28,29,25)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 1, 1 $ $289$ $4$ $( 2, 5,17,14)( 3, 9,16,10)( 4,13,15, 6)( 7, 8,12,11)(19,22,34,31)(20,26,33,27) (21,30,32,23)(24,25,29,28)$
$ 8, 8, 8, 8, 1, 1 $ $289$ $8$ $( 2, 9,14, 3,17,10, 5,16)( 4, 8, 6, 7,15,11,13,12)(19,26,31,20,34,27,22,33) (21,25,23,24,32,28,30,29)$
$ 8, 8, 8, 8, 1, 1 $ $289$ $8$ $( 2,10,14,16,17, 9, 5, 3)( 4,11, 6,12,15, 8,13, 7)(19,27,31,33,34,26,22,20) (21,28,23,29,32,25,30,24)$
$ 8, 8, 8, 8, 1, 1 $ $289$ $8$ $( 2, 3, 5, 9,17,16,14,10)( 4, 7,13, 8,15,12, 6,11)(19,20,22,26,34,33,31,27) (21,24,30,25,32,29,23,28)$
$ 8, 8, 8, 8, 1, 1 $ $289$ $8$ $( 2,16, 5,10,17, 3,14, 9)( 4,12,13,11,15, 7, 6, 8)(19,33,22,27,34,20,31,26) (21,29,30,28,32,24,23,25)$
$ 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 $ $34$ $2$ $(19,34)(20,33)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)$
$ 17, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ $272$ $34$ $( 1, 4, 7,10,13,16, 2, 5, 8,11,14,17, 3, 6, 9,12,15)(18,28)(19,27)(20,26) (21,25)(22,24)(29,34)(30,33)(31,32)$
$ 17, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ $272$ $34$ $( 1,10, 2,11, 3,12, 4,13, 5,14, 6,15, 7,16, 8,17, 9)(18,31)(19,30)(20,29) (21,28)(22,27)(23,26)(24,25)(32,34)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 1, 1 $ $578$ $4$ $( 2,14,17, 5)( 3,10,16, 9)( 4, 6,15,13)( 7,11,12, 8)(19,22,34,31)(20,26,33,27) (21,30,32,23)(24,25,29,28)$
$ 8, 8, 8, 8, 1, 1 $ $578$ $8$ $( 2, 9,14, 3,17,10, 5,16)( 4, 8, 6, 7,15,11,13,12)(19,27,31,33,34,26,22,20) (21,28,23,29,32,25,30,24)$
$ 8, 8, 8, 8, 1, 1 $ $578$ $8$ $( 2, 3, 5, 9,17,16,14,10)( 4, 7,13, 8,15,12, 6,11)(19,33,22,27,34,20,31,26) (21,29,30,28,32,24,23,25)$
$ 8, 8, 8, 8, 2 $ $578$ $8$ $( 1,25,17,19,13,29,14,18)( 2,31, 4,26,12,23,10,28)( 3,20, 8,33,11,34, 6,21) ( 5,32,16,30, 9,22,15,24)( 7,27)$
$ 8, 8, 8, 8, 2 $ $578$ $8$ $( 1,24, 2,34,15,28,14,18)( 3,27,11,22,13,25, 5,30)( 4,20, 7,33,12,32, 9,19) ( 6,23,16,21,10,29,17,31)( 8,26)$
$ 34 $ $272$ $34$ $( 1,23, 5,28, 9,33,13,21,17,26, 4,31, 8,19,12,24,16,29, 3,34, 7,22,11,27,15, 32, 2,20, 6,25,10,30,14,18)$
$ 34 $ $272$ $34$ $( 1,19, 4,27, 7,18,10,26,13,34,16,25, 2,33, 5,24, 8,32,11,23,14,31,17,22, 3, 30, 6,21, 9,29,12,20,15,28)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $34$ $2$ $( 1,24)( 2,21)( 3,18)( 4,32)( 5,29)( 6,26)( 7,23)( 8,20)( 9,34)(10,31)(11,28) (12,25)(13,22)(14,19)(15,33)(16,30)(17,27)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 2 $ $578$ $4$ $( 1,32,14,18)( 2,27,13,23)( 3,22,12,28)( 4,34,11,33)( 5,29,10,21)( 6,24, 9,26) ( 7,19, 8,31)(15,30,17,20)(16,25)$
$ 8, 8, 8, 8, 2 $ $578$ $8$ $( 1,28, 2,22,15,29,14,18)( 3,33,11,19,13,24, 5,21)( 4,27, 7,26,12,30, 9,31) ( 6,32,16,23,10,25,17,34)( 8,20)$
$ 8, 8, 8, 8, 2 $ $578$ $8$ $( 1,29,17,22,13,28,14,18)( 2,19, 4,33,12,21,10,24)( 3,26, 8,27,11,31, 6,30) ( 5,23,16,32, 9,34,15,25)( 7,20)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 2 $ $578$ $4$ $( 1,30,14,18)( 2,33,13,32)( 3,19,12,29)( 4,22,11,26)( 5,25,10,23)( 6,28, 9,20) ( 7,31, 8,34)(15,21,17,27)(16,24)$
$ 34 $ $272$ $34$ $( 1,21, 5,24, 9,27,13,30,17,33, 4,19, 8,22,12,25,16,28, 3,31, 7,34,11,20,15, 23, 2,26, 6,29,10,32,14,18)$
$ 34 $ $272$ $34$ $( 1,19, 4,34, 7,32,10,30,13,28,16,26, 2,24, 5,22, 8,20,11,18,14,33,17,31, 3, 29, 6,27, 9,25,12,23,15,21)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $34$ $2$ $( 1,30)( 2,18)( 3,23)( 4,28)( 5,33)( 6,21)( 7,26)( 8,31)( 9,19)(10,24)(11,29) (12,34)(13,22)(14,27)(15,32)(16,20)(17,25)$

Group invariants

Order:  $9248=2^{5} \cdot 17^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  Data not available
Character table: Data not available.