Properties

Label 34T22
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 $t$ :  $22$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,16,17,8,4,6,5,14)(2,7,13,10,3,15,9,12)(18,21,28,33,22,19,29,24)(23,27,25,26,34,30,32,31), (1,26,10,31,9,21,11,24,7,18,15,30,16,23,14,20)(2,19,8,28,13,27,3,29,6,25,17,33,12,34,5,32)(4,22)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
4:  $C_4$
8:  $C_8$
16:  $C_{16}$
272:  $F_{17}$ x 2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 17: None

Low degree siblings

34T22 x 7

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

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.