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

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.