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

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.