Properties

Label 34T49
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 $t$ :  $49$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,15,16,10,12,17,4,14,5,8,7,13,11,6,2,9)(18,31,19,34,28,27,24,32,22,26,21,23,29,30,33,25), (1,25,9,31,7,21,16,32)(2,30,13,34,6,33,12,29)(3,18,17,20,5,28,8,26)(4,23)(10,19,11,24,15,27,14,22)
$|\Aut(F/K)|$:  $1$

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:  $D_{4}$, $C_4\times C_2$, $Q_8$
16:  $D_{8}$, $C_4:C_4$, $Q_{16}$
32:  32T48
64:  32T376

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 $ $64$ $17$ $( 1,10, 2,11, 3,12, 4,13, 5,14, 6,15, 7,16, 8,17, 9)(18,26,34,25,33,24,32,23, 31,22,30,21,29,20,28,19,27)$
$ 17, 17 $ $64$ $17$ $( 1,11, 4,14, 7,17,10, 3,13, 6,16, 9, 2,12, 5,15, 8)(18,25,32,22,29,19,26,33, 23,30,20,27,34,24,31,21,28)$
$ 17, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $32$ $17$ $(18,32,29,26,23,20,34,31,28,25,22,19,33,30,27,24,21)$
$ 17, 17 $ $64$ $17$ $( 1,10, 2,11, 3,12, 4,13, 5,14, 6,15, 7,16, 8,17, 9)(18,23,28,33,21,26,31,19, 24,29,34,22,27,32,20,25,30)$
$ 17, 17 $ $64$ $17$ $( 1, 9,17, 8,16, 7,15, 6,14, 5,13, 4,12, 3,11, 2,10)(18,24,30,19,25,31,20,26, 32,21,27,33,22,28,34,23,29)$
$ 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 $ $578$ $4$ $( 2, 5,17,14)( 3, 9,16,10)( 4,13,15, 6)( 7, 8,12,11)(19,31,34,22)(20,27,33,26) (21,23,32,30)(24,28,29,25)$
$ 8, 8, 8, 8, 1, 1 $ $578$ $8$ $( 2, 3, 5, 9,17,16,14,10)( 4, 7,13, 8,15,12, 6,11)(19,27,31,33,34,26,22,20) (21,28,23,29,32,25,30,24)$
$ 8, 8, 8, 8, 1, 1 $ $578$ $8$ $( 2,16, 5,10,17, 3,14, 9)( 4,12,13,11,15, 7, 6, 8)(19,26,31,20,34,27,22,33) (21,25,23,24,32,28,30,29)$
$ 16, 16, 1, 1 $ $578$ $16$ $( 2,12, 3, 6, 5,11, 9, 4,17, 7,16,13,14, 8,10,15)(19,21,27,28,31,23,33,29,34, 32,26,25,22,30,20,24)$
$ 16, 16, 1, 1 $ $578$ $16$ $( 2, 7, 3,13, 5, 8, 9,15,17,12,16, 6,14,11,10, 4)(19,32,27,25,31,30,33,24,34, 21,26,28,22,23,20,29)$
$ 16, 16, 1, 1 $ $578$ $16$ $( 2,11,16,15, 5, 7,10, 6,17, 8, 3, 4,14,12, 9,13)(19,23,26,24,31,32,20,28,34, 30,27,29,22,21,33,25)$
$ 16, 16, 1, 1 $ $578$ $16$ $( 2, 8,16, 4, 5,12,10,13,17,11, 3,15,14, 7, 9, 6)(19,30,26,29,31,21,20,25,34, 23,27,24,22,32,33,28)$
$ 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 $ $544$ $34$ $( 1,10, 2,11, 3,12, 4,13, 5,14, 6,15, 7,16, 8,17, 9)(18,26)(19,25)(20,24) (21,23)(27,34)(28,33)(29,32)(30,31)$
$ 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, 1, 1 $ $578$ $8$ $( 2, 3, 5, 9,17,16,14,10)( 4, 7,13, 8,15,12, 6,11)(19,26,31,20,34,27,22,33) (21,25,23,24,32,28,30,29)$
$ 8, 8, 8, 8, 1, 1 $ $578$ $8$ $( 2,16, 5,10,17, 3,14, 9)( 4,12,13,11,15, 7, 6, 8)(19,27,31,33,34,26,22,20) (21,28,23,29,32,25,30,24)$
$ 16, 16, 1, 1 $ $578$ $16$ $( 2,12, 3, 6, 5,11, 9, 4,17, 7,16,13,14, 8,10,15)(19,32,27,25,31,30,33,24,34, 21,26,28,22,23,20,29)$
$ 16, 16, 1, 1 $ $578$ $16$ $( 2, 7, 3,13, 5, 8, 9,15,17,12,16, 6,14,11,10, 4)(19,21,27,28,31,23,33,29,34, 32,26,25,22,30,20,24)$
$ 16, 16, 1, 1 $ $578$ $16$ $( 2,11,16,15, 5, 7,10, 6,17, 8, 3, 4,14,12, 9,13)(19,30,26,29,31,21,20,25,34, 23,27,24,22,32,33,28)$
$ 16, 16, 1, 1 $ $578$ $16$ $( 2, 8,16, 4, 5,12,10,13,17,11, 3,15,14, 7, 9, 6)(19,23,26,24,31,32,20,28,34, 30,27,29,22,21,33,25)$
$ 8, 8, 8, 8, 2 $ $2312$ $8$ $( 1,25, 9,31, 7,21,16,32)( 2,30,13,34, 6,33,12,29)( 3,18,17,20, 5,28, 8,26) ( 4,23)(10,19,11,24,15,27,14,22)$
$ 8, 8, 8, 8, 2 $ $2312$ $8$ $( 1,22,15,28,10,21,13,32)( 2,20,11,19, 9,23,17,24)( 3,18, 7,27, 8,25, 4,33) ( 5,31,16,26, 6,29,12,34)(14,30)$
$ 8, 8, 8, 8, 2 $ $2312$ $8$ $( 1,28, 8,27,14,31, 7,32)( 2,23, 4,30,13,19,11,29)( 3,18,17,33,12,24,15,26) ( 5,25, 9,22,10,34, 6,20)(16,21)$
$ 8, 8, 8, 8, 2 $ $2312$ $8$ $( 1,31,16,27, 8,28,10,32)( 2,33, 3,18, 7,26, 6,24)( 4,20,11,34, 5,22,15,25) ( 9,30,14,23,17,29,12,19)(13,21)$

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.