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

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.