Properties

Label 34T27
Degree $34$
Order $8160$
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)
$|\Aut(F/K)|$:  $2$
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)

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

Group invariants

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