Properties

Label 34T19
Order \(4624\)
n \(34\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No

Learn more about

Group action invariants

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

Low degree resolvents

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

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 $ $8$ $17$ $( 1, 9,17, 8,16, 7,15, 6,14, 5,13, 4,12, 3,11, 2,10)(18,20,22,24,26,28,30,32, 34,19,21,23,25,27,29,31,33)$
$ 17, 17 $ $8$ $17$ $( 1, 8,15, 5,12, 2, 9,16, 6,13, 3,10,17, 7,14, 4,11)(18,24,30,19,25,31,20,26, 32,21,27,33,22,28,34,23,29)$
$ 17, 17 $ $8$ $17$ $( 1, 5, 9,13,17, 4, 8,12,16, 3, 7,11,15, 2, 6,10,14)(18,19,20,21,22,23,24,25, 26,27,28,29,30,31,32,33,34)$
$ 17, 17 $ $8$ $17$ $( 1,13, 8, 3,15,10, 5,17,12, 7, 2,14, 9, 4,16,11, 6)(18,21,24,27,30,33,19,22, 25,28,31,34,20,23,26,29,32)$
$ 17, 17 $ $8$ $17$ $( 1, 3, 5, 7, 9,11,13,15,17, 2, 4, 6, 8,10,12,14,16)(18,27,19,28,20,29,21,30, 22,31,23,32,24,33,25,34,26)$
$ 17, 17 $ $8$ $17$ $( 1, 7,13, 2, 8,14, 3, 9,15, 4,10,16, 5,11,17, 6,12)(18,28,21,31,24,34,27,20, 30,23,33,26,19,29,22,32,25)$
$ 17, 17 $ $8$ $17$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17)(18,31,27,23,19,32,28,24, 20,33,29,25,21,34,30,26,22)$
$ 17, 17 $ $8$ $17$ $( 1, 4, 7,10,13,16, 2, 5, 8,11,14,17, 3, 6, 9,12,15)(18,23,28,33,21,26,31,19, 24,29,34,22,27,32,20,25,30)$
$ 17, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $16$ $17$ $(18,21,24,27,30,33,19,22,25,28,31,34,20,23,26,29,32)$
$ 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 $ $16$ $17$ $( 1, 8,15, 5,12, 2, 9,16, 6,13, 3,10,17, 7,14, 4,11)(18,27,19,28,20,29,21,30, 22,31,23,32,24,33,25,34,26)$
$ 17, 17 $ $8$ $17$ $( 1,13, 8, 3,15,10, 5,17,12, 7, 2,14, 9, 4,16,11, 6)(18,24,30,19,25,31,20,26, 32,21,27,33,22,28,34,23,29)$
$ 17, 17 $ $16$ $17$ $( 1, 3, 5, 7, 9,11,13,15,17, 2, 4, 6, 8,10,12,14,16)(18,30,25,20,32,27,22,34, 29,24,19,31,26,21,33,28,23)$
$ 17, 17 $ $16$ $17$ $( 1, 7,13, 2, 8,14, 3, 9,15, 4,10,16, 5,11,17, 6,12)(18,31,27,23,19,32,28,24, 20,33,29,25,21,34,30,26,22)$
$ 17, 17 $ $8$ $17$ $( 1,10, 2,11, 3,12, 4,13, 5,14, 6,15, 7,16, 8,17, 9)(18,19,20,21,22,23,24,25, 26,27,28,29,30,31,32,33,34)$
$ 17, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $16$ $17$ $(18,27,19,28,20,29,21,30,22,31,23,32,24,33,25,34,26)$
$ 17, 17 $ $16$ $17$ $( 1, 9,17, 8,16, 7,15, 6,14, 5,13, 4,12, 3,11, 2,10)(18,29,23,34,28,22,33,27, 21,32,26,20,31,25,19,30,24)$
$ 17, 17 $ $8$ $17$ $( 1, 3, 5, 7, 9,11,13,15,17, 2, 4, 6, 8,10,12,14,16)(18,19,20,21,22,23,24,25, 26,27,28,29,30,31,32,33,34)$
$ 17, 17 $ $16$ $17$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17)(18,23,28,33,21,26,31,19, 24,29,34,22,27,32,20,25,30)$
$ 17, 17 $ $8$ $17$ $( 1,11, 4,14, 7,17,10, 3,13, 6,16, 9, 2,12, 5,15, 8)(18,21,24,27,30,33,19,22, 25,28,31,34,20,23,26,29,32)$
$ 17, 17 $ $16$ $17$ $( 1, 9,17, 8,16, 7,15, 6,14, 5,13, 4,12, 3,11, 2,10)(18,30,25,20,32,27,22,34, 29,24,19,31,26,21,33,28,23)$
$ 17, 17 $ $8$ $17$ $( 1, 7,13, 2, 8,14, 3, 9,15, 4,10,16, 5,11,17, 6,12)(18,21,24,27,30,33,19,22, 25,28,31,34,20,23,26,29,32)$
$ 17, 17 $ $8$ $17$ $( 1,14,10, 6, 2,15,11, 7, 3,16,12, 8, 4,17,13, 9, 5)(18,27,19,28,20,29,21,30, 22,31,23,32,24,33,25,34,26)$
$ 17, 17 $ $16$ $17$ $( 1, 8,15, 5,12, 2, 9,16, 6,13, 3,10,17, 7,14, 4,11)(18,20,22,24,26,28,30,32, 34,19,21,23,25,27,29,31,33)$
$ 17, 17 $ $8$ $17$ $( 1, 6,11,16, 4, 9,14, 2, 7,12,17, 5,10,15, 3, 8,13)(18,28,21,31,24,34,27,20, 30,23,33,26,19,29,22,32,25)$
$ 17, 17 $ $8$ $17$ $( 1,16,14,12,10, 8, 6, 4, 2,17,15,13,11, 9, 7, 5, 3)(18,31,27,23,19,32,28,24, 20,33,29,25,21,34,30,26,22)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $289$ $2$ $( 2,17)( 3,16)( 4,15)( 5,14)( 6,13)( 7,12)( 8,11)( 9,10)(19,34)(20,33)(21,32) (22,31)(23,30)(24,29)(25,28)(26,27)$
$ 34 $ $136$ $34$ $( 1,32, 9,34,17,19, 8,21,16,23, 7,25,15,27, 6,29,14,31, 5,33,13,18, 4,20,12, 22, 3,24,11,26, 2,28,10,30)$
$ 34 $ $136$ $34$ $( 1,34, 8,23,15,29, 5,18,12,24, 2,30, 9,19,16,25, 6,31,13,20, 3,26,10,32,17, 21, 7,27,14,33, 4,22,11,28)$
$ 34 $ $136$ $34$ $( 1,21, 6,18,11,32,16,29, 4,26, 9,23,14,20, 2,34, 7,31,12,28,17,25, 5,22,10, 19,15,33, 3,30, 8,27,13,24)$
$ 34 $ $136$ $34$ $( 1,33,17,20,16,24,15,28,14,32,13,19,12,23,11,27,10,31, 9,18, 8,22, 7,26, 6, 30, 5,34, 4,21, 3,25, 2,29)$
$ 34 $ $136$ $34$ $( 1,18,16,26,14,34,12,25,10,33, 8,24, 6,32, 4,23, 2,31,17,22,15,30,13,21,11, 29, 9,20, 7,28, 5,19, 3,27)$
$ 34 $ $136$ $34$ $( 1,25, 4,30, 7,18,10,23,13,28,16,33, 2,21, 5,26, 8,31,11,19,14,24,17,29, 3, 34, 6,22, 9,27,12,32,15,20)$
$ 34 $ $136$ $34$ $( 1,26,12,33, 6,23,17,30,11,20, 5,27,16,34,10,24, 4,31,15,21, 9,28, 3,18,14, 25, 8,32, 2,22,13,29, 7,19)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $68$ $2$ $( 1,31)( 2,27)( 3,23)( 4,19)( 5,32)( 6,28)( 7,24)( 8,20)( 9,33)(10,29)(11,25) (12,21)(13,34)(14,30)(15,26)(16,22)(17,18)$
$ 34 $ $136$ $34$ $( 1,23, 5,24, 9,25,13,26,17,27, 4,28, 8,29,12,30,16,31, 3,32, 7,33,11,34,15, 18, 2,19, 6,20,10,21,14,22)$
$ 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 $ $578$ $8$ $( 2,16, 5,10,17, 3,14, 9)( 4,12,13,11,15, 7, 6, 8)(19,26,31,20,34,27,22,33) (21,25,23,24,32,28,30,29)$
$ 8, 8, 8, 8, 1, 1 $ $578$ $8$ $( 2, 3, 5, 9,17,16,14,10)( 4, 7,13, 8,15,12, 6,11)(19,27,31,33,34,26,22,20) (21,28,23,29,32,25,30,24)$
$ 34 $ $136$ $34$ $( 1,32,10,19, 2,23,11,27, 3,31,12,18, 4,22,13,26, 5,30,14,34, 6,21,15,25, 7, 29,16,33, 8,20,17,24, 9,28)$
$ 34 $ $136$ $34$ $( 1,34,14,19,10,21, 6,23, 2,25,15,27,11,29, 7,31, 3,33,16,18,12,20, 8,22, 4, 24,17,26,13,28, 9,30, 5,32)$
$ 34 $ $136$ $34$ $( 1,33,12,19, 6,22,17,25,11,28, 5,31,16,34,10,20, 4,23,15,26, 9,29, 3,32,14, 18, 8,21, 2,24,13,27, 7,30)$
$ 34 $ $136$ $34$ $( 1,18,16,19,14,20,12,21,10,22, 8,23, 6,24, 4,25, 2,26,17,27,15,28,13,29,11, 30, 9,31, 7,32, 5,33, 3,34)$
$ 34 $ $136$ $34$ $( 1,24,11,19, 4,31,14,26, 7,21,17,33,10,28, 3,23,13,18, 6,30,16,25, 9,20, 2, 32,12,27, 5,22,15,34, 8,29)$
$ 34 $ $136$ $34$ $( 1,25,13,19, 8,30, 3,24,15,18,10,29, 5,23,17,34,12,28, 7,22, 2,33,14,27, 9, 21, 4,32,16,26,11,20, 6,31)$
$ 34 $ $136$ $34$ $( 1,28, 2,19, 3,27, 4,18, 5,26, 6,34, 7,25, 8,33, 9,24,10,32,11,23,12,31,13, 22,14,30,15,21,16,29,17,20)$
$ 34 $ $136$ $34$ $( 1,26,15,19,12,29, 9,22, 6,32, 3,25,17,18,14,28,11,21, 8,31, 5,24, 2,34,16, 27,13,20,10,30, 7,23, 4,33)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $68$ $2$ $( 1,19)( 2,27)( 3,18)( 4,26)( 5,34)( 6,25)( 7,33)( 8,24)( 9,32)(10,23)(11,31) (12,22)(13,30)(14,21)(15,29)(16,20)(17,28)$

Group invariants

Order:  $4624=2^{4} \cdot 17^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  Data not available
Character table: Data not available.