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Properties

Label 34T43
Degree $34$
Order $16320$
Cyclic no
Abelian no
Solvable no
Primitive no
$p$-group no
Group: $C_2\times \SOMinus(4,4)$

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Show commands: Magma

magma: G := TransitiveGroup(34, 43);
 

Group action invariants

Degree $n$:  $34$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $43$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_2\times \SOMinus(4,4)$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $2$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
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)
magma: Generators(G);
 

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $16320=2^{6} \cdot 3 \cdot 5 \cdot 17$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  no
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  16320.f
magma: IdentifyGroup(G);
 
Character table: not available.

magma: CharacterTable(G);