Properties

Label 34T38
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 $t$ :  $38$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,5,7,8,17,13,11,10)(2,14,3,6,16,4,15,12)(18,28,31,20,32,22,19,30)(21,34,26,27,29,33,24,23), (1,30,5,20)(2,19,4,31)(3,25)(6,26,17,24)(7,32,16,18)(8,21,15,29)(9,27,14,23)(10,33,13,34)(11,22,12,28), (1,21,14,27,10,33,6,22,2,28,15,34,11,23,7,29,3,18,16,24,12,30,8,19,4,25,17,31,13,20,9,26,5,32)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 7
4:  $C_2^2$ x 7
8:  $D_{4}$ x 2, $C_2^3$
16:  $D_4\times C_2$
32:  16T44

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, 17 $ $16$ $17$ $( 1, 8,15, 5,12, 2, 9,16, 6,13, 3,10,17, 7,14, 4,11)(18,27,19,28,20,29,21,30, 22,31,23,32,24,33,25,34,26)$
$ 17, 17 $ $16$ $17$ $( 1, 5, 9,13,17, 4, 8,12,16, 3, 7,11,15, 2, 6,10,14)(18,28,21,31,24,34,27,20, 30,23,33,26,19,29,22,32,25)$
$ 17, 17 $ $16$ $17$ $( 1,13, 8, 3,15,10, 5,17,12, 7, 2,14, 9, 4,16,11, 6)(18,31,27,23,19,32,28,24, 20,33,29,25,21,34,30,26,22)$
$ 17, 17 $ $16$ $17$ $( 1, 3, 5, 7, 9,11,13,15,17, 2, 4, 6, 8,10,12,14,16)(18,23,28,33,21,26,31,19, 24,29,34,22,27,32,20,25,30)$
$ 17, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $16$ $17$ $(18,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19)$
$ 17, 17 $ $32$ $17$ $( 1, 5, 9,13,17, 4, 8,12,16, 3, 7,11,15, 2, 6,10,14)(18,27,19,28,20,29,21,30, 22,31,23,32,24,33,25,34,26)$
$ 17, 17 $ $32$ $17$ $( 1,13, 8, 3,15,10, 5,17,12, 7, 2,14, 9, 4,16,11, 6)(18,30,25,20,32,27,22,34, 29,24,19,31,26,21,33,28,23)$
$ 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 $ $16$ $17$ $( 1, 4, 7,10,13,16, 2, 5, 8,11,14,17, 3, 6, 9,12,15)(18,33,31,29,27,25,23,21, 19,34,32,30,28,26,24,22,20)$
$ 17, 17 $ $16$ $17$ $( 1,16,14,12,10, 8, 6, 4, 2,17,15,13,11, 9, 7, 5, 3)(18,29,23,34,28,22,33,27, 21,32,26,20,31,25,19,30,24)$
$ 17, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $16$ $17$ $(18,32,29,26,23,20,34,31,28,25,22,19,33,30,27,24,21)$
$ 17, 17 $ $32$ $17$ $( 1, 3, 5, 7, 9,11,13,15,17, 2, 4, 6, 8,10,12,14,16)(18,20,22,24,26,28,30,32, 34,19,21,23,25,27,29,31,33)$
$ 17, 17 $ $16$ $17$ $( 1,10, 2,11, 3,12, 4,13, 5,14, 6,15, 7,16, 8,17, 9)(18,29,23,34,28,22,33,27, 21,32,26,20,31,25,19,30,24)$
$ 17, 17 $ $16$ $17$ $( 1,11, 4,14, 7,17,10, 3,13, 6,16, 9, 2,12, 5,15, 8)(18,34,33,32,31,30,29,28, 27,26,25,24,23,22,21,20,19)$
$ 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 $ $578$ $4$ $( 2,14,17, 5)( 3,10,16, 9)( 4, 6,15,13)( 7,11,12, 8)(19,22,34,31)(20,26,33,27) (21,30,32,23)(24,25,29,28)$
$ 8, 8, 8, 8, 1, 1 $ $578$ $8$ $( 2,10,14,16,17, 9, 5, 3)( 4,11, 6,12,15, 8,13, 7)(19,20,22,26,34,33,31,27) (21,24,30,25,32,29,23,28)$
$ 8, 8, 8, 8, 1, 1 $ $578$ $8$ $( 2, 9,14, 3,17,10, 5,16)( 4, 8, 6, 7,15,11,13,12)(19,33,22,27,34,20,31,26) (21,29,30,28,32,24,23,25)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 2 $ $1156$ $4$ $( 1,30, 5,20)( 2,19, 4,31)( 3,25)( 6,26,17,24)( 7,32,16,18)( 8,21,15,29) ( 9,27,14,23)(10,33,13,34)(11,22,12,28)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 2 $ $1156$ $4$ $( 1,25, 2,20)( 3,32,17,30)( 4,27,16,18)( 5,22,15,23)( 6,34,14,28)( 7,29,13,33) ( 8,24,12,21)( 9,19,11,26)(10,31)$
$ 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 $ $272$ $34$ $( 1, 8,15, 5,12, 2, 9,16, 6,13, 3,10,17, 7,14, 4,11)(18,27)(19,26)(20,25) (21,24)(22,23)(28,34)(29,33)(30,32)$
$ 17, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ $272$ $34$ $( 1, 5, 9,13,17, 4, 8,12,16, 3, 7,11,15, 2, 6,10,14)(18,28)(19,27)(20,26) (21,25)(22,24)(29,34)(30,33)(31,32)$
$ 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 $ $578$ $8$ $( 2,10,14,16,17, 9, 5, 3)( 4,11, 6,12,15, 8,13, 7)(19,33,22,27,34,20,31,26) (21,29,30,28,32,24,23,25)$
$ 8, 8, 8, 8, 1, 1 $ $578$ $8$ $( 2, 9,14, 3,17,10, 5,16)( 4, 8, 6, 7,15,11,13,12)(19,20,22,26,34,33,31,27) (21,24,30,25,32,29,23,28)$
$ 34 $ $272$ $34$ $( 1,30, 9,27,17,24, 8,21,16,18, 7,32,15,29, 6,26,14,23, 5,20,13,34, 4,31,12, 28, 3,25,11,22, 2,19,10,33)$
$ 34 $ $272$ $34$ $( 1,29, 6,25,11,21,16,34, 4,30, 9,26,14,22, 2,18, 7,31,12,27,17,23, 5,19,10, 32,15,28, 3,24, 8,20,13,33)$
$ 34 $ $272$ $34$ $( 1,27,17,21,16,32,15,26,14,20,13,31,12,25,11,19,10,30, 9,24, 8,18, 7,29, 6, 23, 5,34, 4,28, 3,22, 2,33)$
$ 34 $ $272$ $34$ $( 1,25,11,34, 4,26,14,18, 7,27,17,19,10,28, 3,20,13,29, 6,21,16,30, 9,22, 2, 31,12,23, 5,32,15,24, 8,33)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $68$ $2$ $( 1,33)( 2,22)( 3,28)( 4,34)( 5,23)( 6,29)( 7,18)( 8,24)( 9,30)(10,19)(11,25) (12,31)(13,20)(14,26)(15,32)(16,21)(17,27)$
$ 34 $ $272$ $34$ $( 1,25, 6,34,11,26,16,18, 4,27, 9,19,14,28, 2,20, 7,29,12,21,17,30, 5,22,10, 31,15,23, 3,32, 8,24,13,33)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $68$ $2$ $( 1,34)( 2,29)( 3,24)( 4,19)( 5,31)( 6,26)( 7,21)( 8,33)( 9,28)(10,23)(11,18) (12,30)(13,25)(14,20)(15,32)(16,27)(17,22)$
$ 34 $ $272$ $34$ $( 1,18, 8,34,15,33, 5,32,12,31, 2,30, 9,29,16,28, 6,27,13,26, 3,25,10,24,17, 23, 7,22,14,21, 4,20,11,19)$
$ 34 $ $272$ $34$ $( 1,23, 9,34,17,28, 8,22,16,33, 7,27,15,21, 6,32,14,26, 5,20,13,31, 4,25,12, 19, 3,30,11,24, 2,18,10,29)$
$ 34 $ $272$ $34$ $( 1,22, 2,34, 3,29, 4,24, 5,19, 6,31, 7,26, 8,21, 9,33,10,28,11,23,12,18,13, 30,14,25,15,20,16,32,17,27)$

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.