Properties

Label 34T50
Order \(18496\)
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 $t$ :  $50$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,30,4,26,11,28,16,27,5,19,2,23,12,21,7,22)(3,33)(6,29,10,18,8,32,9,25,17,20,13,31,15,34,14,24), (1,5,15,6,9,8,14,12,7,3,10,2,16,17,11,13)(18,22,33,25,20,19,29,31,28,24,30,21,26,27,34,32)
$|\Aut(F/K)|$:  $1$

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:  $D_{4}$ x 2, $C_8$ x 2, $C_4\times C_2$
16:  $C_8:C_2$, $C_2^2:C_4$, $C_8\times C_2$
32:  $C_2^2 : C_8$
64:  32T271

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 $ $64$ $17$ $( 1, 7,13, 2, 8,14, 3, 9,15, 4,10,16, 5,11,17, 6,12)(18,33,31,29,27,25,23,21, 19,34,32,30,28,26,24,22,20)$
$ 17, 17 $ $64$ $17$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17)(18,29,23,34,28,22,33,27, 21,32,26,20,31,25,19,30,24)$
$ 17, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $32$ $17$ $(18,31,27,23,19,32,28,24,20,33,29,25,21,34,30,26,22)$
$ 17, 17 $ $32$ $17$ $( 1, 7,13, 2, 8,14, 3, 9,15, 4,10,16, 5,11,17, 6,12)(18,29,23,34,28,22,33,27, 21,32,26,20,31,25,19,30,24)$
$ 17, 17 $ $32$ $17$ $( 1, 4, 7,10,13,16, 2, 5, 8,11,14,17, 3, 6, 9,12,15)(18,30,25,20,32,27,22,34, 29,24,19,31,26,21,33,28,23)$
$ 17, 17 $ $64$ $17$ $( 1,17,16,15,14,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(18,20,22,24,26,28,30,32, 34,19,21,23,25,27,29,31,33)$
$ 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 $ $1156$ $16$ $( 1,30, 4,26,11,28,16,27, 5,19, 2,23,12,21, 7,22)( 3,33)( 6,29,10,18, 8,32, 9, 25,17,20,13,31,15,34,14,24)$
$ 16, 16, 2 $ $1156$ $16$ $( 1,21,11,29,16,33,10,18, 7,19,14,28, 9,24,15,22)( 2,32, 3,26,12,23, 8,30, 6, 25, 5,31,13,34,17,27)( 4,20)$
$ 16, 16, 2 $ $1156$ $16$ $( 1,29, 5,26,13,20,12,25,10,18, 6,21,15,27,16,22)( 2,24, 7,33,17,34, 3,19, 9, 23, 4,31,11,30, 8,28)(14,32)$
$ 16, 16, 2 $ $1156$ $16$ $( 1,25,16,19, 3,31,12,24,11,21,13,27, 9,32,17,22)( 2,28,14,30, 7,26, 4,34,10, 18,15,33, 5,20, 8,29)( 6,23)$
$ 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 $ $544$ $34$ $( 1, 7,13, 2, 8,14, 3, 9,15, 4,10,16, 5,11,17, 6,12)(18,33)(19,32)(20,31) (21,30)(22,29)(23,28)(24,27)(25,26)$
$ 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 $ $578$ $8$ $( 2, 9,14, 3,17,10, 5,16)( 4, 8, 6, 7,15,11,13,12)(19,27,31,33,34,26,22,20) (21,28,23,29,32,25,30,24)$
$ 8, 8, 8, 8, 1, 1 $ $578$ $8$ $( 2, 3, 5, 9,17,16,14,10)( 4, 7,13, 8,15,12, 6,11)(19,33,22,27,34,20,31,26) (21,29,30,28,32,24,23,25)$
$ 16, 16, 1, 1 $ $578$ $16$ $( 2,12, 3, 6, 5,11, 9, 4,17, 7,16,13,14, 8,10,15)(19,25,33,21,22,29,27,30,34, 28,20,32,31,24,26,23)$
$ 16, 16, 1, 1 $ $578$ $16$ $( 2, 7, 3,13, 5, 8, 9,15,17,12,16, 6,14,11,10, 4)(19,28,33,32,22,24,27,23,34, 25,20,21,31,29,26,30)$
$ 16, 16, 1, 1 $ $578$ $16$ $( 2, 8,16, 4, 5,12,10,13,17,11, 3,15,14, 7, 9, 6)(19,24,20,30,22,25,26,32,34, 29,33,23,31,28,27,21)$
$ 16, 16, 1, 1 $ $578$ $16$ $( 2,11,16,15, 5, 7,10, 6,17, 8, 3, 4,14,12, 9,13)(19,29,20,23,22,28,26,21,34, 24,33,30,31,25,27,32)$
$ 16, 16, 1, 1 $ $578$ $16$ $( 2, 4,10,11,14, 6,16,12,17,15, 9, 8, 5,13, 3, 7)(19,23,26,24,31,32,20,28,34, 30,27,29,22,21,33,25)$
$ 16, 16, 1, 1 $ $578$ $16$ $( 2,15,10, 8,14,13,16, 7,17, 4, 9,11, 5, 6, 3,12)(19,30,26,29,31,21,20,25,34, 23,27,24,22,32,33,28)$
$ 16, 16, 1, 1 $ $578$ $16$ $( 2, 6, 9, 7,14,15, 3,11,17,13,10,12, 5, 4,16, 8)(19,32,27,25,31,30,33,24,34, 21,26,28,22,23,20,29)$
$ 16, 16, 1, 1 $ $578$ $16$ $( 2,13, 9,12,14, 4, 3, 8,17, 6,10, 7, 5,15,16,11)(19,21,27,28,31,23,33,29,34, 32,26,25,22,30,20,24)$
$ 8, 8, 8, 8, 2 $ $1156$ $8$ $( 1,30,14,32,15,23, 2,21)( 3,29, 5,28,13,24,11,25)( 4,20, 9,26,12,33, 7,27) ( 6,19,17,22,10,34,16,31)( 8,18)$
$ 8, 8, 8, 8, 2 $ $1156$ $8$ $( 1,21)( 2,23,14,30,17,19, 5,29)( 3,25,10,22,16,34, 9,20)( 4,27, 6,31,15,32, 13,28)( 7,33,11,24,12,26, 8,18)$
$ 34 $ $544$ $34$ $( 1,29,16,20,14,28,12,19,10,27, 8,18, 6,26, 4,34, 2,25,17,33,15,24,13,32,11, 23, 9,31, 7,22, 5,30, 3,21)$
$ 34 $ $544$ $34$ $( 1,28,15,23,12,18, 9,30, 6,25, 3,20,17,32,14,27,11,22, 8,34, 5,29, 2,24,16, 19,13,31,10,26, 7,21, 4,33)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $68$ $2$ $( 1,31)( 2,27)( 3,23)( 4,19)( 5,32)( 6,28)( 7,24)( 8,20)( 9,33)(10,29)(11,25) (12,21)(13,34)(14,30)(15,26)(16,22)(17,18)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 2 $ $1156$ $4$ $( 1,25, 5,21)( 2,24, 4,22)( 3,23)( 6,20,17,26)( 7,19,16,27)( 8,18,15,28) ( 9,34,14,29)(10,33,13,30)(11,32,12,31)$

Group invariants

Order:  $18496=2^{6} \cdot 17^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  Data not available
Character table: Data not available.