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

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.