Properties

Label 34T27
Order \(8160\)
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 $t$ :  $27$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,25,34,2,26,33)(3,12,5,4,11,6)(7,21,24,8,22,23)(9,10)(13,16,31,14,15,32)(17,18)(19,28,30,20,27,29), (1,28,3,15,32,33,14,29,10,18,11,7,26,22,23,20,5,2,27,4,16,31,34,13,30,9,17,12,8,25,21,24,19,6)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
4080:  $\PSL(2,16)$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

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

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 $ $1$ $1$ $()$
$ 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 $ $272$ $5$ $( 1,10,32,19,11)( 2, 9,31,20,12)( 3,34,23,14,17)( 4,33,24,13,18) ( 7,28,22,29,15)( 8,27,21,30,16)$
$ 10, 10, 10, 2, 2 $ $272$ $10$ $( 1, 9,32,20,11, 2,10,31,19,12)( 3,33,23,13,17, 4,34,24,14,18)( 5, 6) ( 7,27,22,30,15, 8,28,21,29,16)(25,26)$
$ 5, 5, 5, 5, 5, 5, 1, 1, 1, 1 $ $272$ $5$ $( 1,19,10,11,32)( 2,20, 9,12,31)( 3,14,34,17,23)( 4,13,33,18,24) ( 7,29,28,15,22)( 8,30,27,16,21)$
$ 10, 10, 10, 2, 2 $ $272$ $10$ $( 1,20,10,12,32, 2,19, 9,11,31)( 3,13,34,18,23, 4,14,33,17,24)( 5, 6) ( 7,30,28,16,22, 8,29,27,15,21)(25,26)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1 $ $272$ $3$ $( 1, 8,23)( 2, 7,24)( 3,19,30)( 4,20,29)( 9,28,13)(10,27,14)(11,16,34) (12,15,33)(17,32,21)(18,31,22)$
$ 6, 6, 6, 6, 6, 2, 2 $ $272$ $6$ $( 1, 7,23, 2, 8,24)( 3,20,30, 4,19,29)( 5, 6)( 9,27,13,10,28,14) (11,15,34,12,16,33)(17,31,21,18,32,22)(25,26)$
$ 15, 15, 1, 1, 1, 1 $ $272$ $15$ $( 1, 3,27,11,17, 8,19,14,16,32,23,30,10,34,21)( 2, 4,28,12,18, 7,20,13,15,31, 24,29, 9,33,22)$
$ 30, 2, 2 $ $272$ $30$ $( 1, 4,27,12,17, 7,19,13,16,31,23,29,10,33,21, 2, 3,28,11,18, 8,20,14,15,32, 24,30, 9,34,22)( 5, 6)(25,26)$
$ 30, 2, 2 $ $272$ $30$ $( 1,29,14,12,21,24,19,28,34,31, 8, 4,10,15,17, 2,30,13,11,22,23,20,27,33,32, 7, 3, 9,16,18)( 5, 6)(25,26)$
$ 15, 15, 1, 1, 1, 1 $ $272$ $15$ $( 1,30,14,11,21,23,19,27,34,32, 8, 3,10,16,17)( 2,29,13,12,22,24,20,28,33,31, 7, 4, 9,15,18)$
$ 15, 15, 1, 1, 1, 1 $ $272$ $15$ $( 1,14,21,19,34, 8,10,17,30,11,23,27,32, 3,16)( 2,13,22,20,33, 7, 9,18,29,12, 24,28,31, 4,15)$
$ 30, 2, 2 $ $272$ $30$ $( 1,13,21,20,34, 7,10,18,30,12,23,28,32, 4,16, 2,14,22,19,33, 8, 9,17,29,11, 24,27,31, 3,15)( 5, 6)(25,26)$
$ 15, 15, 1, 1, 1, 1 $ $272$ $15$ $( 1,27,17,19,16,23,10,21, 3,11, 8,14,32,30,34)( 2,28,18,20,15,24, 9,22, 4,12, 7,13,31,29,33)$
$ 30, 2, 2 $ $272$ $30$ $( 1,28,17,20,16,24,10,22, 3,12, 8,13,32,29,34, 2,27,18,19,15,23, 9,21, 4,11, 7,14,31,30,33)( 5, 6)(25,26)$
$ 17, 17 $ $240$ $17$ $( 1, 8,11,32,23,19,16,26, 5,21, 3,10,34,17,27,14,30)( 2, 7,12,31,24,20,15,25, 6,22, 4, 9,33,18,28,13,29)$
$ 34 $ $240$ $34$ $( 1, 7,11,31,23,20,16,25, 5,22, 3, 9,34,18,27,13,30, 2, 8,12,32,24,19,15,26, 6,21, 4,10,33,17,28,14,29)$
$ 17, 17 $ $240$ $17$ $( 1,23, 5,34,30,32,26,10,14,11,16, 3,27, 8,19,21,17)( 2,24, 6,33,29,31,25, 9, 13,12,15, 4,28, 7,20,22,18)$
$ 34 $ $240$ $34$ $( 1,24, 5,33,30,31,26, 9,14,12,16, 4,27, 7,19,22,17, 2,23, 6,34,29,32,25,10, 13,11,15, 3,28, 8,20,21,18)$
$ 34 $ $240$ $34$ $( 1,12,23,15, 5, 4,34,28,30, 7,32,20,26,22,10,18,14, 2,11,24,16, 6, 3,33,27, 29, 8,31,19,25,21, 9,17,13)$
$ 17, 17 $ $240$ $17$ $( 1,11,23,16, 5, 3,34,27,30, 8,32,19,26,21,10,17,14)( 2,12,24,15, 6, 4,33,28, 29, 7,31,20,25,22, 9,18,13)$
$ 34 $ $240$ $34$ $( 1, 6,30,25,14,15,27,20,17,24,34,31,10,12, 3, 7,21, 2, 5,29,26,13,16,28,19, 18,23,33,32, 9,11, 4, 8,22)$
$ 17, 17 $ $240$ $17$ $( 1, 5,30,26,14,16,27,19,17,23,34,32,10,11, 3, 8,21)( 2, 6,29,25,13,15,28,20, 18,24,33,31, 9,12, 4, 7,22)$
$ 17, 17 $ $240$ $17$ $( 1,16,34, 8,26,17,11, 5,27,32,21,14,23, 3,30,19,10)( 2,15,33, 7,25,18,12, 6, 28,31,22,13,24, 4,29,20, 9)$
$ 34 $ $240$ $34$ $( 1,15,34, 7,26,18,11, 6,27,31,21,13,23, 4,30,20,10, 2,16,33, 8,25,17,12, 5, 28,32,22,14,24, 3,29,19, 9)$
$ 34 $ $240$ $34$ $( 1,25,27,24,10, 7, 5,13,19,33,11,22,30,15,17,31, 3, 2,26,28,23, 9, 8, 6,14, 20,34,12,21,29,16,18,32, 4)$
$ 17, 17 $ $240$ $17$ $( 1,26,27,23,10, 8, 5,14,19,34,11,21,30,16,17,32, 3)( 2,25,28,24, 9, 7, 6,13, 20,33,12,22,29,15,18,31, 4)$
$ 17, 17 $ $240$ $17$ $( 1,34,26,11,27,21,23,30,10,16, 8,17, 5,32,14, 3,19)( 2,33,25,12,28,22,24,29, 9,15, 7,18, 6,31,13, 4,20)$
$ 34 $ $240$ $34$ $( 1,33,26,12,27,22,23,29,10,15, 8,18, 5,31,14, 4,19, 2,34,25,11,28,21,24,30, 9,16, 7,17, 6,32,13, 3,20)$
$ 34 $ $240$ $34$ $( 1,28,10, 6,19,12,30,18, 3,25,23, 7,14,33,21,15,32, 2,27, 9, 5,20,11,29,17, 4,26,24, 8,13,34,22,16,31)$
$ 17, 17 $ $240$ $17$ $( 1,27,10, 5,19,11,30,17, 3,26,23, 8,14,34,21,16,32)( 2,28, 9, 6,20,12,29,18, 4,25,24, 7,13,33,22,15,31)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $255$ $2$ $( 1,11)( 2,12)( 3,16)( 4,15)( 5,27)( 6,28)( 7,25)( 8,26)( 9,33)(10,34)(13,18) (14,17)(19,21)(20,22)(23,32)(24,31)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $255$ $2$ $( 1,12)( 2,11)( 3,15)( 4,16)( 5,28)( 6,27)( 7,26)( 8,25)( 9,34)(10,33)(13,17) (14,18)(19,22)(20,21)(23,31)(24,32)(29,30)$

Group invariants

Order:  $8160=2^{5} \cdot 3 \cdot 5 \cdot 17$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  Data not available
Character table: Data not available.