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

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.