Properties

Label 34T43
Order \(16320\)
n \(34\)
Cyclic No
Abelian No
Solvable No
Primitive No
$p$-group No

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Group action invariants

Degree $n$ :  $34$
Transitive number $t$ :  $43$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,9,12,34,5,31,7,26,4,29,28,21,19,13,16,2,10,11,33,6,32,8,25,3,30,27,22,20,14,15)(17,18)(23,24), (1,21,12,6)(2,22,11,5)(3,32,13,17)(4,31,14,18)(7,27,10,34)(8,28,9,33)(15,24,20,30)(16,23,19,29)(25,26)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
8160:  $\PSL(2,16):C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 17: $\PSL(2,16):C_2$

Low degree siblings

34T43

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 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)(29,30)(31,32)(33,34)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $68$ $2$ $( 1,27)( 2,28)( 3, 7)( 4, 8)( 5, 6)( 9,10)(11,30)(12,29)(13,16)(14,15)(17,31) (18,32)(19,20)(21,25)(22,26)(23,24)(33,34)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $68$ $2$ $( 1,28)( 2,27)( 3, 8)( 4, 7)(11,29)(12,30)(13,15)(14,16)(17,32)(18,31)(21,26) (22,25)$
$ 6, 6, 6, 6, 6, 2, 2 $ $272$ $6$ $( 1,29, 7, 2,30, 8)( 3,28,11, 4,27,12)( 5,23,10, 6,24, 9)(13,32,26,14,31,25) (15,17,21,16,18,22)(19,20)(33,34)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1 $ $272$ $3$ $( 1,30, 7)( 2,29, 8)( 3,27,11)( 4,28,12)( 5,24,10)( 6,23, 9)(13,31,26) (14,32,25)(15,18,21)(16,17,22)$
$ 6, 6, 6, 6, 3, 3, 1, 1, 1, 1 $ $1360$ $6$ $( 1, 4,30,28, 7,12)( 2, 3,29,27, 8,11)( 5,10,24)( 6, 9,23)(13,21,31,15,26,18) (14,22,32,16,25,17)$
$ 6, 6, 6, 6, 6, 2, 2 $ $1360$ $6$ $( 1, 3,30,27, 7,11)( 2, 4,29,28, 8,12)( 5, 9,24, 6,10,23)(13,22,31,16,26,17) (14,21,32,15,25,18)(19,20)(33,34)$
$ 17, 17 $ $480$ $17$ $( 1,33,16,17,30,25,10, 4,28,19, 5,14,22,24, 7,12,32)( 2,34,15,18,29,26, 9, 3, 27,20, 6,13,21,23, 8,11,31)$
$ 34 $ $480$ $34$ $( 1,34,16,18,30,26,10, 3,28,20, 5,13,22,23, 7,11,32, 2,33,15,17,29,25, 9, 4, 27,19, 6,14,21,24, 8,12,31)$
$ 34 $ $480$ $34$ $( 1,15,30, 9,28, 6,22, 8,32,34,17,26, 4,20,14,23,12, 2,16,29,10,27, 5,21, 7, 31,33,18,25, 3,19,13,24,11)$
$ 17, 17 $ $480$ $17$ $( 1,16,30,10,28, 5,22, 7,32,33,17,25, 4,19,14,24,12)( 2,15,29, 9,27, 6,21, 8, 31,34,18,26, 3,20,13,23,11)$
$ 34 $ $480$ $34$ $( 1, 9,22,34, 4,23,16,27, 7,18,19,11,30, 6,32,26,14, 2,10,21,33, 3,24,15,28, 8,17,20,12,29, 5,31,25,13)$
$ 17, 17 $ $480$ $17$ $( 1,10,22,33, 4,24,16,28, 7,17,19,12,30, 5,32,25,14)( 2, 9,21,34, 3,23,15,27, 8,18,20,11,29, 6,31,26,13)$
$ 17, 17 $ $480$ $17$ $( 1,22, 4,16, 7,19,30,32,14,10,33,24,28,17,12, 5,25)( 2,21, 3,15, 8,20,29,31, 13, 9,34,23,27,18,11, 6,26)$
$ 34 $ $480$ $34$ $( 1,21, 4,15, 7,20,30,31,14, 9,33,23,28,18,12, 6,25, 2,22, 3,16, 8,19,29,32, 13,10,34,24,27,17,11, 5,26)$
$ 5, 5, 5, 5, 5, 5, 1, 1, 1, 1 $ $272$ $5$ $( 1, 5,22,16,10)( 2, 6,21,15, 9)( 3,29,23, 8,11)( 4,30,24, 7,12) (13,26,20,31,34)(14,25,19,32,33)$
$ 10, 10, 10, 2, 2 $ $272$ $10$ $( 1, 6,22,15,10, 2, 5,21,16, 9)( 3,30,23, 7,11, 4,29,24, 8,12)(13,25,20,32,34, 14,26,19,31,33)(17,18)(27,28)$
$ 5, 5, 5, 5, 5, 5, 1, 1, 1, 1 $ $272$ $5$ $( 1,16, 5,10,22)( 2,15, 6, 9,21)( 3, 8,29,11,23)( 4, 7,30,12,24) (13,31,26,34,20)(14,32,25,33,19)$
$ 10, 10, 10, 2, 2 $ $272$ $10$ $( 1,15, 5, 9,22, 2,16, 6,10,21)( 3, 7,29,12,23, 4, 8,30,11,24)(13,32,26,33,20, 14,31,25,34,19)(17,18)(27,28)$
$ 10, 10, 10, 2, 2 $ $816$ $10$ $( 1,26, 5,20,22,31,16,34,10,13)( 2,25, 6,19,21,32,15,33, 9,14)( 3, 7,29,12,23, 4, 8,30,11,24)(17,27)(18,28)$
$ 10, 10, 5, 5, 2, 2 $ $816$ $10$ $( 1,25, 5,19,22,32,16,33,10,14)( 2,26, 6,20,21,31,15,34, 9,13)( 3, 8,29,11,23) ( 4, 7,30,12,24)(17,28)(18,27)$
$ 10, 10, 5, 5, 2, 2 $ $816$ $10$ $( 1,33,22,25,10,32, 5,14,16,19)( 2,34,21,26, 9,31, 6,13,15,20)( 3,29,23, 8,11) ( 4,30,24, 7,12)(17,28)(18,27)$
$ 10, 10, 10, 2, 2 $ $816$ $10$ $( 1,34,22,26,10,31, 5,13,16,20)( 2,33,21,25, 9,32, 6,14,15,19)( 3,30,23, 7,11, 4,29,24, 8,12)(17,27)(18,28)$
$ 30, 2, 2 $ $544$ $30$ $( 1,26,16, 3, 5,31,17,13,33,21,10,23,30,27, 7, 2,25,15, 4, 6,32,18,14,34,22, 9,24,29,28, 8)(11,12)(19,20)$
$ 15, 15, 1, 1, 1, 1 $ $544$ $15$ $( 1,25,16, 4, 5,32,17,14,33,22,10,24,30,28, 7)( 2,26,15, 3, 6,31,18,13,34,21, 9,23,29,27, 8)$
$ 15, 15, 1, 1, 1, 1 $ $544$ $15$ $( 1,14, 7,17,28,32,30, 5,24, 4,10,16,22,25,33)( 2,13, 8,18,27,31,29, 6,23, 3, 9,15,21,26,34)$
$ 30, 2, 2 $ $544$ $30$ $( 1,13, 7,18,28,31,30, 6,24, 3,10,15,22,26,33, 2,14, 8,17,27,32,29, 5,23, 4, 9,16,21,25,34)(11,12)(19,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $255$ $2$ $( 1,32)( 2,31)( 3,23)( 4,24)( 5,17)( 6,18)( 9,15)(10,16)(11,29)(12,30)(13,21) (14,22)(19,25)(20,26)(27,34)(28,33)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $255$ $2$ $( 1,31)( 2,32)( 3,24)( 4,23)( 5,18)( 6,17)( 7, 8)( 9,16)(10,15)(11,30)(12,29) (13,22)(14,21)(19,26)(20,25)(27,33)(28,34)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 2 $ $1020$ $4$ $( 1,29,32,11)( 2,30,31,12)( 3,22,23,14)( 4,21,24,13)( 5,26,17,20)( 6,25,18,19) ( 7, 8)( 9,28,15,33)(10,27,16,34)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 1, 1 $ $1020$ $4$ $( 1,30,32,12)( 2,29,31,11)( 3,21,23,13)( 4,22,24,14)( 5,25,17,19)( 6,26,18,20) ( 9,27,15,34)(10,28,16,33)$

Group invariants

Order:  $16320=2^{6} \cdot 3 \cdot 5 \cdot 17$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  Data not available
Character table: Data not available.