Properties

Label 34T48
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 $t$ :  $48$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,25,7,31)(2,26,6,30)(3,27,5,29)(4,28)(8,32,17,24)(9,33,16,23)(10,34,15,22)(11,18,14,21)(12,19,13,20), (1,10,4,8,11,9,16,17,5,13,2,15,12,14,7,6)(18,27,22,21,31,33,30,26,32,23,28,29,19,34,20,24), (1,30,7,31,13,32,2,33,8,34,14,18,3,19,9,20,15,21,4,22,10,23,16,24,5,25,11,26,17,27,6,28,12,29)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 7
4:  $C_2^2$ x 7
8:  $D_{4}$ x 2, $C_2^3$
16:  $D_{8}$ x 2, $D_4\times C_2$
32:  16T29
64:  32T303

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 $ $32$ $17$ $( 1, 3, 5, 7, 9,11,13,15,17, 2, 4, 6, 8,10,12,14,16)(18,25,32,22,29,19,26,33, 23,30,20,27,34,24,31,21,28)$
$ 17, 17 $ $32$ $17$ $( 1, 7,13, 2, 8,14, 3, 9,15, 4,10,16, 5,11,17, 6,12)(18,22,26,30,34,21,25,29, 33,20,24,28,32,19,23,27,31)$
$ 17, 17 $ $32$ $17$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17)(18,30,25,20,32,27,22,34, 29,24,19,31,26,21,33,28,23)$
$ 17, 17 $ $32$ $17$ $( 1, 4, 7,10,13,16, 2, 5, 8,11,14,17, 3, 6, 9,12,15)(18,20,22,24,26,28,30,32, 34,19,21,23,25,27,29,31,33)$
$ 17, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $32$ $17$ $(18,30,25,20,32,27,22,34,29,24,19,31,26,21,33,28,23)$
$ 17, 17 $ $32$ $17$ $( 1, 3, 5, 7, 9,11,13,15,17, 2, 4, 6, 8,10,12,14,16)(18,20,22,24,26,28,30,32, 34,19,21,23,25,27,29,31,33)$
$ 17, 17 $ $32$ $17$ $( 1, 4, 7,10,13,16, 2, 5, 8,11,14,17, 3, 6, 9,12,15)(18,32,29,26,23,20,34,31, 28,25,22,19,33,30,27,24,21)$
$ 17, 17 $ $32$ $17$ $( 1, 5, 9,13,17, 4, 8,12,16, 3, 7,11,15, 2, 6,10,14)(18,27,19,28,20,29,21,30, 22,31,23,32,24,33,25,34,26)$
$ 17, 17 $ $32$ $17$ $( 1, 3, 5, 7, 9,11,13,15,17, 2, 4, 6, 8,10,12,14,16)(18,27,19,28,20,29,21,30, 22,31,23,32,24,33,25,34,26)$
$ 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 $ $578$ $4$ $( 2,14,17, 5)( 3,10,16, 9)( 4, 6,15,13)( 7,11,12, 8)(19,22,34,31)(20,26,33,27) (21,30,32,23)(24,25,29,28)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 2 $ $2312$ $4$ $( 1,25, 7,31)( 2,26, 6,30)( 3,27, 5,29)( 4,28)( 8,32,17,24)( 9,33,16,23) (10,34,15,22)(11,18,14,21)(12,19,13,20)$
$ 8, 8, 8, 8, 1, 1 $ $578$ $8$ $( 2,10,14,16,17, 9, 5, 3)( 4,11, 6,12,15, 8,13, 7)(19,20,22,26,34,33,31,27) (21,24,30,25,32,29,23,28)$
$ 8, 8, 8, 8, 1, 1 $ $578$ $8$ $( 2, 9,14, 3,17,10, 5,16)( 4, 8, 6, 7,15,11,13,12)(19,33,22,27,34,20,31,26) (21,29,30,28,32,24,23,25)$
$ 16, 16, 1, 1 $ $578$ $16$ $( 2, 6, 9, 7,14,15, 3,11,17,13,10,12, 5, 4,16, 8)(19,25,33,21,22,29,27,30,34, 28,20,32,31,24,26,23)$
$ 16, 16, 1, 1 $ $578$ $16$ $( 2,13, 9,12,14, 4, 3, 8,17, 6,10, 7, 5,15,16,11)(19,28,33,32,22,24,27,23,34, 25,20,21,31,29,26,30)$
$ 16, 16, 1, 1 $ $578$ $16$ $( 2,15,10, 8,14,13,16, 7,17, 4, 9,11, 5, 6, 3,12)(19,29,20,23,22,28,26,21,34, 24,33,30,31,25,27,32)$
$ 16, 16, 1, 1 $ $578$ $16$ $( 2, 4,10,11,14, 6,16,12,17,15, 9, 8, 5,13, 3, 7)(19,24,20,30,22,25,26,32,34, 29,33,23,31,28,27,21)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 2 $ $2312$ $4$ $( 1,25,16,32)( 2,30,15,27)( 3,18,14,22)( 4,23,13,34)( 5,28,12,29)( 6,33,11,24) ( 7,21,10,19)( 8,26, 9,31)(17,20)$
$ 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 $ $544$ $34$ $( 1, 3, 5, 7, 9,11,13,15,17, 2, 4, 6, 8,10,12,14,16)(18,25)(19,24)(20,23) (21,22)(26,34)(27,33)(28,32)(29,31)$
$ 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 $ $544$ $34$ $( 1,25, 4,28, 7,31,10,34,13,20,16,23, 2,26, 5,29, 8,32,11,18,14,21,17,24, 3, 27, 6,30, 9,33,12,19,15,22)$
$ 34 $ $544$ $34$ $( 1,29,14,25,10,21, 6,34, 2,30,15,26,11,22, 7,18, 3,31,16,27,12,23, 8,19, 4, 32,17,28,13,24, 9,20, 5,33)$
$ 34 $ $544$ $34$ $( 1,27, 9,18,17,26, 8,34,16,25, 7,33,15,24, 6,32,14,23, 5,31,13,22, 4,30,12, 21, 3,29,11,20, 2,28,10,19)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $136$ $2$ $( 1,34)( 2,18)( 3,19)( 4,20)( 5,21)( 6,22)( 7,23)( 8,24)( 9,25)(10,26)(11,27) (12,28)(13,29)(14,30)(15,31)(16,32)(17,33)$
$ 34 $ $544$ $34$ $( 1,18,12,29, 6,23,17,34,11,28, 5,22,16,33,10,27, 4,21,15,32, 9,26, 3,20,14, 31, 8,25, 2,19,13,30, 7,24)$
$ 8, 8, 8, 8, 1, 1 $ $578$ $8$ $( 2,10,14,16,17, 9, 5, 3)( 4,11, 6,12,15, 8,13, 7)(19,33,22,27,34,20,31,26) (21,29,30,28,32,24,23,25)$
$ 8, 8, 8, 8, 1, 1 $ $578$ $8$ $( 2, 9,14, 3,17,10, 5,16)( 4, 8, 6, 7,15,11,13,12)(19,20,22,26,34,33,31,27) (21,24,30,25,32,29,23,28)$
$ 16, 16, 1, 1 $ $578$ $16$ $( 2, 6, 9, 7,14,15, 3,11,17,13,10,12, 5, 4,16, 8)(19,28,33,32,22,24,27,23,34, 25,20,21,31,29,26,30)$
$ 16, 16, 1, 1 $ $578$ $16$ $( 2,13, 9,12,14, 4, 3, 8,17, 6,10, 7, 5,15,16,11)(19,25,33,21,22,29,27,30,34, 28,20,32,31,24,26,23)$
$ 16, 16, 1, 1 $ $578$ $16$ $( 2,15,10, 8,14,13,16, 7,17, 4, 9,11, 5, 6, 3,12)(19,24,20,30,22,25,26,32,34, 29,33,23,31,28,27,21)$
$ 16, 16, 1, 1 $ $578$ $16$ $( 2, 4,10,11,14, 6,16,12,17,15, 9, 8, 5,13, 3, 7)(19,29,20,23,22,28,26,21,34, 24,33,30,31,25,27,32)$
$ 34 $ $544$ $34$ $( 1,25,12,29, 6,33,17,20,11,24, 5,28,16,32,10,19, 4,23,15,27, 9,31, 3,18,14, 22, 8,26, 2,30,13,34, 7,21)$
$ 34 $ $544$ $34$ $( 1,27, 9,33,17,22, 8,28,16,34, 7,23,15,29, 6,18,14,24, 5,30,13,19, 4,25,12, 31, 3,20,11,26, 2,32,10,21)$
$ 34 $ $544$ $34$ $( 1,26, 2,31, 3,19, 4,24, 5,29, 6,34, 7,22, 8,27, 9,32,10,20,11,25,12,30,13, 18,14,23,15,28,16,33,17,21)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $136$ $2$ $( 1,21)( 2,26)( 3,31)( 4,19)( 5,24)( 6,29)( 7,34)( 8,22)( 9,27)(10,32)(11,20) (12,25)(13,30)(14,18)(15,23)(16,28)(17,33)$
$ 34 $ $544$ $34$ $( 1,30,13,22, 8,31, 3,23,15,32,10,24, 5,33,17,25,12,34, 7,26, 2,18,14,27, 9, 19, 4,28,16,20,11,29, 6,21)$

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.