Properties

Label 34T37
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 $t$ :  $37$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,32,12,22,6,29,17,19,11,26,5,33,16,23,10,30,4,20,15,27,9,34,3,24,14,31,8,21,2,28,13,18,7,25), (1,32,12,34)(2,26,11,23)(3,20,10,29)(4,31,9,18)(5,25,8,24)(6,19,7,30)(13,28,17,21)(14,22,16,27)(15,33)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
8:  $D_{4}$
16:  $D_{8}$
32:  16T55

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 $ $16$ $17$ $( 1,12, 6,17,11, 5,16,10, 4,15, 9, 3,14, 8, 2,13, 7)(18,25,32,22,29,19,26,33, 23,30,20,27,34,24,31,21,28)$
$ 17, 17 $ $16$ $17$ $( 1,17,16,15,14,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(18,22,26,30,34,21,25,29, 33,20,24,28,32,19,23,27,31)$
$ 17, 17 $ $16$ $17$ $( 1,15,12, 9, 6, 3,17,14,11, 8, 5, 2,16,13,10, 7, 4)(18,30,25,20,32,27,22,34, 29,24,19,31,26,21,33,28,23)$
$ 17, 17 $ $16$ $17$ $( 1, 9,17, 8,16, 7,15, 6,14, 5,13, 4,12, 3,11, 2,10)(18,20,22,24,26,28,30,32, 34,19,21,23,25,27,29,31,33)$
$ 17, 17 $ $16$ $17$ $( 1, 8,15, 5,12, 2, 9,16, 6,13, 3,10,17, 7,14, 4,11)(18,24,30,19,25,31,20,26, 32,21,27,33,22,28,34,23,29)$
$ 17, 17 $ $16$ $17$ $( 1, 5, 9,13,17, 4, 8,12,16, 3, 7,11,15, 2, 6,10,14)(18,19,20,21,22,23,24,25, 26,27,28,29,30,31,32,33,34)$
$ 17, 17 $ $16$ $17$ $( 1,13, 8, 3,15,10, 5,17,12, 7, 2,14, 9, 4,16,11, 6)(18,21,24,27,30,33,19,22, 25,28,31,34,20,23,26,29,32)$
$ 17, 17 $ $16$ $17$ $( 1, 3, 5, 7, 9,11,13,15,17, 2, 4, 6, 8,10,12,14,16)(18,27,19,28,20,29,21,30, 22,31,23,32,24,33,25,34,26)$
$ 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 $ $32$ $17$ $( 1,12, 6,17,11, 5,16,10, 4,15, 9, 3,14, 8, 2,13, 7)(18,26,34,25,33,24,32,23, 31,22,30,21,29,20,28,19,27)$
$ 17, 17 $ $32$ $17$ $( 1,17,16,15,14,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(18,23,28,33,21,26,31,19, 24,29,34,22,27,32,20,25,30)$
$ 17, 17 $ $32$ $17$ $( 1, 9,17, 8,16, 7,15, 6,14, 5,13, 4,12, 3,11, 2,10)(18,21,24,27,30,33,19,22, 25,28,31,34,20,23,26,29,32)$
$ 17, 17 $ $32$ $17$ $( 1,15,12, 9, 6, 3,17,14,11, 8, 5, 2,16,13,10, 7, 4)(18,33,31,29,27,25,23,21, 19,34,32,30,28,26,24,22,20)$
$ 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 $ $578$ $4$ $( 2, 5,17,14)( 3, 9,16,10)( 4,13,15, 6)( 7, 8,12,11)(19,31,34,22)(20,27,33,26) (21,23,32,30)(24,28,29,25)$
$ 34 $ $272$ $34$ $( 1,32,12,22, 6,29,17,19,11,26, 5,33,16,23,10,30, 4,20,15,27, 9,34, 3,24,14, 31, 8,21, 2,28,13,18, 7,25)$
$ 34 $ $272$ $34$ $( 1,22,17,26,16,30,15,34,14,21,13,25,12,29,11,33,10,20, 9,24, 8,28, 7,32, 6, 19, 5,23, 4,27, 3,31, 2,18)$
$ 34 $ $272$ $34$ $( 1,19,10,34, 2,32,11,30, 3,28,12,26, 4,24,13,22, 5,20,14,18, 6,33,15,31, 7, 29,16,27, 8,25,17,23, 9,21)$
$ 34 $ $272$ $34$ $( 1,27, 6,24,11,21,16,18, 4,32, 9,29,14,26, 2,23, 7,20,12,34,17,31, 5,28,10, 25,15,22, 3,19, 8,33,13,30)$
$ 34 $ $272$ $34$ $( 1,34,11,28, 4,22,14,33, 7,27,17,21,10,32, 3,26,13,20, 6,31,16,25, 9,19, 2, 30,12,24, 5,18,15,29, 8,23)$
$ 34 $ $272$ $34$ $( 1,33, 3,25, 5,34, 7,26, 9,18,11,27,13,19,15,28,17,20, 2,29, 4,21, 6,30, 8, 22,10,31,12,23,14,32,16,24)$
$ 34 $ $272$ $34$ $( 1,28,14,27,10,26, 6,25, 2,24,15,23,11,22, 7,21, 3,20,16,19,12,18, 8,34, 4, 33,17,32,13,31, 9,30, 5,29)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $136$ $2$ $( 1,20)( 2,33)( 3,29)( 4,25)( 5,21)( 6,34)( 7,30)( 8,26)( 9,22)(10,18)(11,31) (12,27)(13,23)(14,19)(15,32)(16,28)(17,24)$
$ 34 $ $272$ $34$ $( 1,26,15,21,12,33, 9,28, 6,23, 3,18,17,30,14,25,11,20, 8,32, 5,27, 2,22,16, 34,13,29,10,24, 7,19, 4,31)$
$ 8, 8, 8, 8, 1, 1 $ $578$ $8$ $( 2, 3, 5, 9,17,16,14,10)( 4, 7,13, 8,15,12, 6,11)(19,27,31,33,34,26,22,20) (21,28,23,29,32,25,30,24)$
$ 8, 8, 8, 8, 1, 1 $ $578$ $8$ $( 2,16, 5,10,17, 3,14, 9)( 4,12,13,11,15, 7, 6, 8)(19,26,31,20,34,27,22,33) (21,25,23,24,32,28,30,29)$
$ 16, 16, 1, 1 $ $578$ $16$ $( 2,11,16,15, 5, 7,10, 6,17, 8, 3, 4,14,12, 9,13)(19,23,26,24,31,32,20,28,34, 30,27,29,22,21,33,25)$
$ 16, 16, 1, 1 $ $578$ $16$ $( 2, 8,16, 4, 5,12,10,13,17,11, 3,15,14, 7, 9, 6)(19,30,26,29,31,21,20,25,34, 23,27,24,22,32,33,28)$
$ 16, 16, 1, 1 $ $578$ $16$ $( 2, 7, 3,13, 5, 8, 9,15,17,12,16, 6,14,11,10, 4)(19,32,27,25,31,30,33,24,34, 21,26,28,22,23,20,29)$
$ 16, 16, 1, 1 $ $578$ $16$ $( 2,12, 3, 6, 5,11, 9, 4,17, 7,16,13,14, 8,10,15)(19,21,27,28,31,23,33,29,34, 32,26,25,22,30,20,24)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 2 $ $2312$ $4$ $( 1,32,15,33)( 2,26,14,22)( 3,20,13,28)( 4,31,12,34)( 5,25,11,23)( 6,19,10,29) ( 7,30, 9,18)( 8,24)(16,27,17,21)$

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.