Properties

Label 34T36
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 $t$ :  $36$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,34,8,30,13,32,2,31,16,23,9,27,4,25,15,26)(3,28,7,33,5,22,6,19,14,29,10,24,12,18,11,21)(17,20), (1,12,9,16,11,17,3,13)(2,4,5,14,10,8,7,15)(18,19,27,23,25,24,33,20)(21,26,32,29,22,34,28,31)
$|\Aut(F/K)|$:  $1$

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

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.