Properties

Label 34T41
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 $t$ :  $41$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,25,2,34,15,32,14,23)(3,26,11,30,13,31,5,27)(4,18,7,28,12,22,9,29)(6,19,16,24,10,21,17,33)(8,20), (1,8,17,14,15,9,11,16,3,13,4,7,6,12,10,5)(18,29,31,19,23,33,24,27,26,32,30,25,21,28,20,34)
$|\Aut(F/K)|$:  $1$

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:  $C_{16}$ x 2, $C_8\times C_2$
32:  32T32
272:  $F_{17}$ x 2
544:  34T9 x 2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 17: None

Low degree siblings

34T41 x 7

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

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.