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

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.