Properties

Label 34T43
Degree $34$
Order $16320$
Cyclic no
Abelian no
Solvable no
Primitive no
$p$-group no

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Group action invariants

Degree $n$:  $34$
Transitive number $t$:  $43$
Parity:  $-1$
Primitive:  no
Nilpotency class:  $-1$ (not nilpotent)
$|\Aut(F/K)|$:  $2$
Generators:  (1,9,12,34,5,31,7,26,4,29,28,21,19,13,16,2,10,11,33,6,32,8,25,3,30,27,22,20,14,15)(17,18)(23,24), (1,21,12,6)(2,22,11,5)(3,32,13,17)(4,31,14,18)(7,27,10,34)(8,28,9,33)(15,24,20,30)(16,23,19,29)(25,26)

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$8160$:  $\PSL(2,16):C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 17: $\PSL(2,16):C_2$

Low degree siblings

34T43

Siblings are shown 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 $ $272$ $6$ $( 1, 6,16, 2, 5,15)( 3,19,13, 4,20,14)( 7,23,17, 8,24,18)( 9,10) (11,25,29,12,26,30)(21,22)(27,33,31,28,34,32)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1 $ $272$ $3$ $( 1, 5,16)( 2, 6,15)( 3,20,13)( 4,19,14)( 7,24,17)( 8,23,18)(11,26,29) (12,25,30)(27,34,31)(28,33,32)$
$ 10, 10, 10, 2, 2 $ $272$ $10$ $( 1,34,30,20, 7, 2,33,29,19, 8)( 3,17,15,28,26, 4,18,16,27,25)( 5,31,12,13,24, 6,32,11,14,23)( 9,10)(21,22)$
$ 5, 5, 5, 5, 5, 5, 1, 1, 1, 1 $ $272$ $5$ $( 1,33,30,19, 7)( 2,34,29,20, 8)( 3,18,15,27,26)( 4,17,16,28,25) ( 5,32,12,14,24)( 6,31,11,13,23)$
$ 10, 10, 10, 2, 2 $ $272$ $10$ $( 1,20,33, 8,30, 2,19,34, 7,29)( 3,28,18,25,15, 4,27,17,26,16)( 5,13,32,23,12, 6,14,31,24,11)( 9,10)(21,22)$
$ 5, 5, 5, 5, 5, 5, 1, 1, 1, 1 $ $272$ $5$ $( 1,19,33, 7,30)( 2,20,34, 8,29)( 3,27,18,26,15)( 4,28,17,25,16) ( 5,14,32,24,12)( 6,13,31,23,11)$
$ 15, 15, 1, 1, 1, 1 $ $544$ $15$ $( 1,25,24,33, 4, 5,30,17,32,19,16,12, 7,28,14)( 2,26,23,34, 3, 6,29,18,31,20, 15,11, 8,27,13)$
$ 30, 2, 2 $ $544$ $30$ $( 1,26,24,34, 4, 6,30,18,32,20,16,11, 7,27,14, 2,25,23,33, 3, 5,29,17,31,19, 15,12, 8,28,13)( 9,10)(21,22)$
$ 30, 2, 2 $ $544$ $30$ $( 1,18,14,29,28, 6, 7, 3,12,34,16,23,19,26,32, 2,17,13,30,27, 5, 8, 4,11,33, 15,24,20,25,31)( 9,10)(21,22)$
$ 15, 15, 1, 1, 1, 1 $ $544$ $15$ $( 1,17,14,30,28, 5, 7, 4,12,33,16,24,19,25,32)( 2,18,13,29,27, 6, 8, 3,11,34, 15,23,20,26,31)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $68$ $2$ $( 1,15)( 2,16)( 3,19)( 4,20)( 5, 6)( 7,18)( 8,17)( 9,22)(10,21)(11,12)(13,14) (23,24)(25,29)(26,30)(27,33)(28,34)(31,32)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $68$ $2$ $( 1,16)( 2,15)( 3,20)( 4,19)( 7,17)( 8,18)( 9,21)(10,22)(25,30)(26,29)(27,34) (28,33)$
$ 10, 10, 10, 2, 2 $ $816$ $10$ $( 1, 3,33,18,30,15,19,27, 7,26)( 2, 4,34,17,29,16,20,28, 8,25)( 5,13,32,23,12, 6,14,31,24,11)( 9,22)(10,21)$
$ 10, 10, 5, 5, 2, 2 $ $816$ $10$ $( 1, 4,33,17,30,16,19,28, 7,25)( 2, 3,34,18,29,15,20,27, 8,26)( 5,14,32,24,12) ( 6,13,31,23,11)( 9,21)(10,22)$
$ 10, 10, 5, 5, 2, 2 $ $816$ $10$ $( 1,28,30, 4, 7,16,33,25,19,17)( 2,27,29, 3, 8,15,34,26,20,18)( 5,32,12,14,24) ( 6,31,11,13,23)( 9,21)(10,22)$
$ 10, 10, 10, 2, 2 $ $816$ $10$ $( 1,27,30, 3, 7,15,33,26,19,18)( 2,28,29, 4, 8,16,34,25,20,17)( 5,31,12,13,24, 6,32,11,14,23)( 9,22)(10,21)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $255$ $2$ $( 1,11)( 2,12)( 3, 7)( 4, 8)( 5, 6)( 9,24)(10,23)(13,16)(14,15)(17,26)(18,25) (19,34)(20,33)(21,32)(22,31)(27,30)(28,29)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $255$ $2$ $( 1,12)( 2,11)( 3, 8)( 4, 7)( 9,23)(10,24)(13,15)(14,16)(17,25)(18,26)(19,33) (20,34)(21,31)(22,32)(27,29)(28,30)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 2 $ $1020$ $4$ $( 1,23,28,20)( 2,24,27,19)( 3,32,13,25)( 4,31,14,26)( 5, 6)( 7,21,16,18) ( 8,22,15,17)( 9,30,34,12)(10,29,33,11)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 1, 1 $ $1020$ $4$ $( 1,24,28,19)( 2,23,27,20)( 3,31,13,26)( 4,32,14,25)( 7,22,16,17)( 8,21,15,18) ( 9,29,34,11)(10,30,33,12)$
$ 6, 6, 6, 6, 6, 2, 2 $ $1360$ $6$ $( 1, 3,28,34,32,13)( 2, 4,27,33,31,14)( 5,29,16, 6,30,15)( 7,11,10,20,22,26) ( 8,12, 9,19,21,25)(17,18)(23,24)$
$ 6, 6, 6, 6, 3, 3, 1, 1, 1, 1 $ $1360$ $6$ $( 1, 4,28,33,32,14)( 2, 3,27,34,31,13)( 5,30,16)( 6,29,15)( 7,12,10,19,22,25) ( 8,11, 9,20,21,26)$
$ 34 $ $480$ $34$ $( 1,31,28,29,22,11,24,13, 7,26,33, 6,19,18, 4,15,10, 2,32,27,30,21,12,23,14, 8,25,34, 5,20,17, 3,16, 9)$
$ 17, 17 $ $480$ $17$ $( 1,32,28,30,22,12,24,14, 7,25,33, 5,19,17, 4,16,10)( 2,31,27,29,21,11,23,13, 8,26,34, 6,20,18, 3,15, 9)$
$ 17, 17 $ $480$ $17$ $( 1,28,22,24, 7,33,19, 4,10,32,30,12,14,25, 5,17,16)( 2,27,21,23, 8,34,20, 3, 9,31,29,11,13,26, 6,18,15)$
$ 34 $ $480$ $34$ $( 1,27,22,23, 7,34,19, 3,10,31,30,11,14,26, 5,18,16, 2,28,21,24, 8,33,20, 4, 9,32,29,12,13,25, 6,17,15)$
$ 34 $ $480$ $34$ $( 1,23,19,31,14,18,28, 8, 4,29,25,15,22,34,10,11, 5, 2,24,20,32,13,17,27, 7, 3,30,26,16,21,33, 9,12, 6)$
$ 17, 17 $ $480$ $17$ $( 1,24,19,32,14,17,28, 7, 4,30,25,16,22,33,10,12, 5)( 2,23,20,31,13,18,27, 8, 3,29,26,15,21,34, 9,11, 6)$
$ 17, 17 $ $480$ $17$ $( 1,19,14,28, 4,25,22,10, 5,24,32,17, 7,30,16,33,12)( 2,20,13,27, 3,26,21, 9, 6,23,31,18, 8,29,15,34,11)$
$ 34 $ $480$ $34$ $( 1,20,14,27, 4,26,22, 9, 5,23,32,18, 7,29,16,34,12, 2,19,13,28, 3,25,21,10, 6,24,31,17, 8,30,15,33,11)$

Group invariants

Order:  $16320=2^{6} \cdot 3 \cdot 5 \cdot 17$
Cyclic:  no
Abelian:  no
Solvable:  no
GAP id:  not available
Character table: not available.