Properties

Label 34T25
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 $t$ :  $25$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,32,12,34,5,25,11,23)(2,26,16,27,4,31,7,30)(3,20)(6,19,15,33,17,21,8,24)(9,18,10,29,14,22,13,28), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,17)(14,16)(18,20,22,24,26,28,30,32,34,19,21,23,25,27,29,31,33)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_4$ x 2, $C_2^2$
8:  $C_4\times C_2$
16:  $C_8:C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 17: None

Low degree siblings

34T26 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 $ $16$ $17$ $( 1, 7,13, 2, 8,14, 3, 9,15, 4,10,16, 5,11,17, 6,12)(18,27,19,28,20,29,21,30, 22,31,23,32,24,33,25,34,26)$
$ 17, 17 $ $16$ $17$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17)(18,28,21,31,24,34,27,20, 30,23,33,26,19,29,22,32,25)$
$ 17, 17 $ $16$ $17$ $( 1, 4, 7,10,13,16, 2, 5, 8,11,14,17, 3, 6, 9,12,15)(18,31,27,23,19,32,28,24, 20,33,29,25,21,34,30,26,22)$
$ 17, 17 $ $16$ $17$ $( 1,10, 2,11, 3,12, 4,13, 5,14, 6,15, 7,16, 8,17, 9)(18,23,28,33,21,26,31,19, 24,29,34,22,27,32,20,25,30)$
$ 17, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $17$ $(18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34)$
$ 17, 17 $ $16$ $17$ $( 1, 7,13, 2, 8,14, 3, 9,15, 4,10,16, 5,11,17, 6,12)(18,28,21,31,24,34,27,20, 30,23,33,26,19,29,22,32,25)$
$ 17, 17 $ $16$ $17$ $( 1, 4, 7,10,13,16, 2, 5, 8,11,14,17, 3, 6, 9,12,15)(18,32,29,26,23,20,34,31, 28,25,22,19,33,30,27,24,21)$
$ 17, 17 $ $16$ $17$ $( 1,10, 2,11, 3,12, 4,13, 5,14, 6,15, 7,16, 8,17, 9)(18,24,30,19,25,31,20,26, 32,21,27,33,22,28,34,23,29)$
$ 17, 17 $ $16$ $17$ $( 1,11, 4,14, 7,17,10, 3,13, 6,16, 9, 2,12, 5,15, 8)(18,34,33,32,31,30,29,28, 27,26,25,24,23,22,21,20,19)$
$ 17, 17 $ $16$ $17$ $( 1,14,10, 6, 2,15,11, 7, 3,16,12, 8, 4,17,13, 9, 5)(18,30,25,20,32,27,22,34, 29,24,19,31,26,21,33,28,23)$
$ 17, 17 $ $16$ $17$ $( 1,16,14,12,10, 8, 6, 4, 2,17,15,13,11, 9, 7, 5, 3)(18,33,31,29,27,25,23,21, 19,34,32,30,28,26,24,22,20)$
$ 17, 17 $ $16$ $17$ $( 1,17,16,15,14,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(18,26,34,25,33,24,32,23, 31,22,30,21,29,20,28,19,27)$
$ 17, 17 $ $16$ $17$ $( 1,15,12, 9, 6, 3,17,14,11, 8, 5, 2,16,13,10, 7, 4)(18,23,28,33,21,26,31,19, 24,29,34,22,27,32,20,25,30)$
$ 17, 17 $ $16$ $17$ $( 1, 8,15, 5,12, 2, 9,16, 6,13, 3,10,17, 7,14, 4,11)(18,21,24,27,30,33,19,22, 25,28,31,34,20,23,26,29,32)$
$ 17, 17 $ $16$ $17$ $( 1,13, 8, 3,15,10, 5,17,12, 7, 2,14, 9, 4,16,11, 6)(18,20,22,24,26,28,30,32, 34,19,21,23,25,27,29,31,33)$
$ 17, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $17$ $(18,21,24,27,30,33,19,22,25,28,31,34,20,23,26,29,32)$
$ 17, 17 $ $16$ $17$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17)(18,31,27,23,19,32,28,24, 20,33,29,25,21,34,30,26,22)$
$ 17, 17 $ $16$ $17$ $( 1,15,12, 9, 6, 3,17,14,11, 8, 5, 2,16,13,10, 7, 4)(18,25,32,22,29,19,26,33, 23,30,20,27,34,24,31,21,28)$
$ 17, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $17$ $(18,27,19,28,20,29,21,30,22,31,23,32,24,33,25,34,26)$
$ 17, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $17$ $(18,28,21,31,24,34,27,20,30,23,33,26,19,29,22,32,25)$
$ 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, 2 $ $578$ $8$ $( 1,32,12,34, 5,25,11,23)( 2,26,16,27, 4,31, 7,30)( 3,20)( 6,19,15,33,17,21, 8,24)( 9,18,10,29,14,22,13,28)$
$ 8, 8, 8, 8, 2 $ $578$ $8$ $( 1,23)( 2,33,14,34,17,30, 5,29)( 3,26,10,28,16,20, 9,18)( 4,19, 6,22,15,27, 13,24)( 7,32,11,21,12,31, 8,25)$
$ 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 $ $136$ $34$ $( 1, 7,13, 2, 8,14, 3, 9,15, 4,10,16, 5,11,17, 6,12)(18,27)(19,26)(20,25) (21,24)(22,23)(28,34)(29,33)(30,32)$
$ 17, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ $136$ $34$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17)(18,28)(19,27)(20,26) (21,25)(22,24)(29,34)(30,33)(31,32)$
$ 17, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ $136$ $34$ $( 1, 4, 7,10,13,16, 2, 5, 8,11,14,17, 3, 6, 9,12,15)(18,31)(19,30)(20,29) (21,28)(22,27)(23,26)(24,25)(32,34)$
$ 17, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ $136$ $34$ $( 1,10, 2,11, 3,12, 4,13, 5,14, 6,15, 7,16, 8,17, 9)(18,23)(19,22)(20,21) (24,34)(25,33)(26,32)(27,31)(28,30)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 1, 1 $ $578$ $4$ $( 2, 5,17,14)( 3, 9,16,10)( 4,13,15, 6)( 7, 8,12,11)(19,31,34,22)(20,27,33,26) (21,23,32,30)(24,28,29,25)$
$ 8, 8, 8, 8, 2 $ $578$ $8$ $( 1,32, 8,24,14,22, 7,30)( 2,26, 4,31,13,28,11,23)( 3,20,17,21,12,34,15,33) ( 5,25, 9,18,10,29, 6,19)(16,27)$
$ 8, 8, 8, 8, 2 $ $578$ $8$ $( 1,23, 5,29, 4,19,17,30)( 2,33, 9,18, 3,26,13,24)( 6,22, 8,25,16,20,14,34) ( 7,32,12,31,15,27,10,28)(11,21)$

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.