Properties

Label 34T32
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 $t$ :  $32$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,20,14,26,10,32,6,21,2,27,15,33,11,22,7,28,3,34,16,23,12,29,8,18,4,24,17,30,13,19,9,25,5,31), (1,15,7,14,10,5,3,9,8,11,2,12,16,4,6,17)(18,32,33,27,29,34,21,31,22,25,24,30,28,23,19,26)
$|\Aut(F/K)|$:  $1$

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:  $D_{16}$

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

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.