Properties

Label 34T40
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 $t$ :  $40$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,13,8,3,15,10,5,17,12,7,2,14,9,4,16,11,6)(18,33,25,27)(19,20,24,23)(21,28,22,32)(26,31,34,29), (1,18,10,21,2,24,11,27,3,30,12,33,4,19,13,22,5,25,14,28,6,31,15,34,7,20,16,23,8,26,17,29,9,32)
$|\Aut(F/K)|$:  $1$

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:  $D_{4}$ x 2, $C_4\times C_2$
16:  $C_2^2:C_4$
32:  $C_4\wr C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 17: None

Low degree siblings

There are no siblings 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $17$ $( 1,15,12, 9, 6, 3,17,14,11, 8, 5, 2,16,13,10, 7, 4)$
$ 17, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $17$ $( 1, 9,17, 8,16, 7,15, 6,14, 5,13, 4,12, 3,11, 2,10)$
$ 17, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $17$ $( 1, 8,15, 5,12, 2, 9,16, 6,13, 3,10,17, 7,14, 4,11)$
$ 17, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $17$ $( 1, 5, 9,13,17, 4, 8,12,16, 3, 7,11,15, 2, 6,10,14)$
$ 17, 17 $ $16$ $17$ $( 1,15,12, 9, 6, 3,17,14,11, 8, 5, 2,16,13,10, 7, 4)(18,34,33,32,31,30,29,28, 27,26,25,24,23,22,21,20,19)$
$ 17, 17 $ $32$ $17$ $( 1, 9,17, 8,16, 7,15, 6,14, 5,13, 4,12, 3,11, 2,10)(18,34,33,32,31,30,29,28, 27,26,25,24,23,22,21,20,19)$
$ 17, 17 $ $32$ $17$ $( 1, 8,15, 5,12, 2, 9,16, 6,13, 3,10,17, 7,14, 4,11)(18,34,33,32,31,30,29,28, 27,26,25,24,23,22,21,20,19)$
$ 17, 17 $ $32$ $17$ $( 1, 5, 9,13,17, 4, 8,12,16, 3, 7,11,15, 2, 6,10,14)(18,34,33,32,31,30,29,28, 27,26,25,24,23,22,21,20,19)$
$ 17, 17 $ $16$ $17$ $( 1, 9,17, 8,16, 7,15, 6,14, 5,13, 4,12, 3,11, 2,10)(18,32,29,26,23,20,34,31, 28,25,22,19,33,30,27,24,21)$
$ 17, 17 $ $32$ $17$ $( 1, 8,15, 5,12, 2, 9,16, 6,13, 3,10,17, 7,14, 4,11)(18,32,29,26,23,20,34,31, 28,25,22,19,33,30,27,24,21)$
$ 17, 17 $ $32$ $17$ $( 1, 5, 9,13,17, 4, 8,12,16, 3, 7,11,15, 2, 6,10,14)(18,32,29,26,23,20,34,31, 28,25,22,19,33,30,27,24,21)$
$ 17, 17 $ $16$ $17$ $( 1, 8,15, 5,12, 2, 9,16, 6,13, 3,10,17, 7,14, 4,11)(18,26,34,25,33,24,32,23, 31,22,30,21,29,20,28,19,27)$
$ 17, 17 $ $32$ $17$ $( 1, 5, 9,13,17, 4, 8,12,16, 3, 7,11,15, 2, 6,10,14)(18,26,34,25,33,24,32,23, 31,22,30,21,29,20,28,19,27)$
$ 17, 17 $ $16$ $17$ $( 1, 5, 9,13,17, 4, 8,12,16, 3, 7,11,15, 2, 6,10,14)(18,25,32,22,29,19,26,33, 23,30,20,27,34,24,31,21,28)$
$ 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 $ $34$ $2$ $(19,34)(20,33)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)$
$ 17, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ $136$ $34$ $( 1,15,12, 9, 6, 3,17,14,11, 8, 5, 2,16,13,10, 7, 4)(19,34)(20,33)(21,32) (22,31)(23,30)(24,29)(25,28)(26,27)$
$ 17, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ $136$ $34$ $( 1, 9,17, 8,16, 7,15, 6,14, 5,13, 4,12, 3,11, 2,10)(19,34)(20,33)(21,32) (22,31)(23,30)(24,29)(25,28)(26,27)$
$ 17, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ $136$ $34$ $( 1, 8,15, 5,12, 2, 9,16, 6,13, 3,10,17, 7,14, 4,11)(19,34)(20,33)(21,32) (22,31)(23,30)(24,29)(25,28)(26,27)$
$ 17, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ $136$ $34$ $( 1, 5, 9,13,17, 4, 8,12,16, 3, 7,11,15, 2, 6,10,14)(19,34)(20,33)(21,32) (22,31)(23,30)(24,29)(25,28)(26,27)$
$ 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $34$ $4$ $(19,22,34,31)(20,26,33,27)(21,30,32,23)(24,25,29,28)$
$ 17, 4, 4, 4, 4, 1 $ $136$ $68$ $( 1,15,12, 9, 6, 3,17,14,11, 8, 5, 2,16,13,10, 7, 4)(19,22,34,31)(20,26,33,27) (21,30,32,23)(24,25,29,28)$
$ 17, 4, 4, 4, 4, 1 $ $136$ $68$ $( 1, 9,17, 8,16, 7,15, 6,14, 5,13, 4,12, 3,11, 2,10)(19,22,34,31)(20,26,33,27) (21,30,32,23)(24,25,29,28)$
$ 17, 4, 4, 4, 4, 1 $ $136$ $68$ $( 1, 8,15, 5,12, 2, 9,16, 6,13, 3,10,17, 7,14, 4,11)(19,22,34,31)(20,26,33,27) (21,30,32,23)(24,25,29,28)$
$ 17, 4, 4, 4, 4, 1 $ $136$ $68$ $( 1, 5, 9,13,17, 4, 8,12,16, 3, 7,11,15, 2, 6,10,14)(19,22,34,31)(20,26,33,27) (21,30,32,23)(24,25,29,28)$
$ 4, 4, 4, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $34$ $4$ $(19,31,34,22)(20,27,33,26)(21,23,32,30)(24,28,29,25)$
$ 17, 4, 4, 4, 4, 1 $ $136$ $68$ $( 1,15,12, 9, 6, 3,17,14,11, 8, 5, 2,16,13,10, 7, 4)(19,31,34,22)(20,27,33,26) (21,23,32,30)(24,28,29,25)$
$ 17, 4, 4, 4, 4, 1 $ $136$ $68$ $( 1, 9,17, 8,16, 7,15, 6,14, 5,13, 4,12, 3,11, 2,10)(19,31,34,22)(20,27,33,26) (21,23,32,30)(24,28,29,25)$
$ 17, 4, 4, 4, 4, 1 $ $136$ $68$ $( 1, 8,15, 5,12, 2, 9,16, 6,13, 3,10,17, 7,14, 4,11)(19,31,34,22)(20,27,33,26) (21,23,32,30)(24,28,29,25)$
$ 17, 4, 4, 4, 4, 1 $ $136$ $68$ $( 1, 5, 9,13,17, 4, 8,12,16, 3, 7,11,15, 2, 6,10,14)(19,31,34,22)(20,27,33,26) (21,23,32,30)(24,28,29,25)$
$ 4, 4, 4, 4, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $578$ $4$ $( 2,17)( 3,16)( 4,15)( 5,14)( 6,13)( 7,12)( 8,11)( 9,10)(19,22,34,31) (20,26,33,27)(21,30,32,23)(24,25,29,28)$
$ 4, 4, 4, 4, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $578$ $4$ $( 2,17)( 3,16)( 4,15)( 5,14)( 6,13)( 7,12)( 8,11)( 9,10)(19,31,34,22) (20,27,33,26)(21,23,32,30)(24,28,29,25)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 1, 1 $ $289$ $4$ $( 2, 5,17,14)( 3, 9,16,10)( 4,13,15, 6)( 7, 8,12,11)(19,22,34,31)(20,26,33,27) (21,30,32,23)(24,25,29,28)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 1, 1 $ $578$ $4$ $( 2, 5,17,14)( 3, 9,16,10)( 4,13,15, 6)( 7, 8,12,11)(19,31,34,22)(20,27,33,26) (21,23,32,30)(24,28,29,25)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 1, 1 $ $289$ $4$ $( 2,14,17, 5)( 3,10,16, 9)( 4, 6,15,13)( 7,11,12, 8)(19,31,34,22)(20,27,33,26) (21,23,32,30)(24,28,29,25)$
$ 34 $ $272$ $34$ $( 1,18,10,21, 2,24,11,27, 3,30,12,33, 4,19,13,22, 5,25,14,28, 6,31,15,34, 7, 20,16,23, 8,26,17,29, 9,32)$
$ 34 $ $272$ $34$ $( 1,18, 7,20,13,22, 2,24, 8,26,14,28, 3,30, 9,32,15,34, 4,19,10,21,16,23, 5, 25,11,27,17,29, 6,31,12,33)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $68$ $2$ $( 1,18)( 2,24)( 3,30)( 4,19)( 5,25)( 6,31)( 7,20)( 8,26)( 9,32)(10,21)(11,27) (12,33)(13,22)(14,28)(15,34)(16,23)(17,29)$
$ 34 $ $272$ $34$ $( 1,18,17,29,16,23,15,34,14,28,13,22,12,33,11,27,10,21, 9,32, 8,26, 7,20, 6, 31, 5,25, 4,19, 3,30, 2,24)$
$ 34 $ $272$ $34$ $( 1,18,13,22, 8,26, 3,30,15,34,10,21, 5,25,17,29,12,33, 7,20, 2,24,14,28, 9, 32, 4,19,16,23,11,27, 6,31)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 2 $ $1156$ $4$ $( 1,18,10,32)( 2,29, 9,21)( 3,23, 8,27)( 4,34, 7,33)( 5,28, 6,22)(11,26,17,24) (12,20,16,30)(13,31,15,19)(14,25)$
$ 8, 8, 8, 8, 2 $ $1156$ $8$ $( 1,18,10,30,12,27, 3,32)( 2,25,14,24,11,20,16,21)( 4,22, 5,29, 9,23, 8,33) ( 6,19,13,34, 7,26,17,28)(15,31)$
$ 8, 8, 8, 8, 2 $ $1156$ $8$ $( 1,18,10,23, 8,20,16,32)( 2,28, 6,34, 7,27, 3,21)( 4,31,15,22, 5,24,11,33) ( 9,30,12,26,17,25,14,29)(13,19)$

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.