Properties

Label 34T24
Order \(4624\)
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 $t$ :  $24$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,23,17,28)(2,18,16,33)(3,30,15,21)(4,25,14,26)(5,20,13,31)(6,32,12,19)(7,27,11,24)(8,22,10,29)(9,34), (1,9,11,3)(2,5,10,7)(4,14,8,15)(12,16,17,13)(18,23,20,32)(21,28,34,27)(22,24,33,31)(25,29,30,26), (1,7,13,2,8,14,3,9,15,4,10,16,5,11,17,6,12)(18,24)(19,23)(20,22)(25,34)(26,33)(27,32)(28,31)(29,30)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 7
4:  $C_2^2$ x 7
8:  $C_2^3$
16:  $Q_8:C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 17: None

Low degree siblings

34T24 x 2

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

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.