Properties

Label 34T34
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 $t$ :  $34$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,32,2,29,11,19,7,31,5,20,4,23,12,33,16,21)(3,26)(6,34,13,30,8,28,14,27,17,18,10,22,15,24,9,25), (1,18)(2,20,14,27,17,33,5,26)(3,22,10,19,16,31,9,34)(4,24,6,28,15,29,13,25)(7,30,11,21,12,23,8,32)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 17: None

Low degree siblings

34T34, 34T36

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

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.