Properties

Label 34T42
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 $t$ :  $42$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,32,15,31,16,20,10,18,12,30,17,26,4,33,14,25,5,22,8,23,7,34,13,19,11,24,6,28,2,21,9,29)(3,27), (1,17,6,4,16,12,2,11,8,9,3,5,10,14,7,15)(18,26,29,28,34,32,27,23,30,22,19,20,31,33,21,25)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
4:  $C_4$
8:  $C_8$
16:  $C_{16}$
32:  32T33

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 17: None

Low degree siblings

34T42 x 8

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

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.