LMFDB - Galois group: 34T19

Show commands: Magma

magma: G := TransitiveGroup(34, 19);

Group action invariants

Degree $n$:	$34$	magma: t, n := TransitiveGroupIdentification(G); n;
Transitive number $t$:	$19$	magma: t, n := TransitiveGroupIdentification(G); t;
Group:	$C_{17}^2:D_8$
Parity:	$-1$	magma: IsEven(G);
Primitive:	no	magma: IsPrimitive(G);
magma: NilpotencyClass(G);
$\card{\Aut(F/K)}$:	$1$	magma: Order(Centralizer(SymmetricGroup(n), G));
Generators:	(1,32,9,34,17,19,8,21,16,23,7,25,15,27,6,29,14,31,5,33,13,18,4,20,12,22,3,24,11,26,2,28,10,30), (1,17,2,15,6,7,5,9)(3,13,10,16,4,11,14,8)(18,34,26,30,28,29,20,33)(19,25,22,32,27,21,24,31)	magma: Generators(G);

Low degree resolvents

|G/N| Galois groups for stem field(s)
$2$: $C_2$ x 3
$4$: $C_2^2$
$8$: $D_{4}$
$16$: $D_{8}$

Resolvents shown for degrees $\leq 47$

\|G/N\|	Galois groups for stem field(s)
$2$:	$C_2$ x 3
$4$:	$C_2^2$
$8$:	$D_{4}$
$16$:	$D_{8}$

Subfields

Degree 2: $C_2$
Degree 17: None

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

Label	Cycle Type	Size	Order	Representative
	$ 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 $	$1$	$1$	$()$
	$ 17, 17 $	$8$	$17$	$( 1, 9,17, 8,16, 7,15, 6,14, 5,13, 4,12, 3,11, 2,10)(18,20,22,24,26,28,30,32, 34,19,21,23,25,27,29,31,33)$
	$ 17, 17 $	$8$	$17$	$( 1, 8,15, 5,12, 2, 9,16, 6,13, 3,10,17, 7,14, 4,11)(18,24,30,19,25,31,20,26, 32,21,27,33,22,28,34,23,29)$
	$ 17, 17 $	$8$	$17$	$( 1, 5, 9,13,17, 4, 8,12,16, 3, 7,11,15, 2, 6,10,14)(18,19,20,21,22,23,24,25, 26,27,28,29,30,31,32,33,34)$
	$ 17, 17 $	$8$	$17$	$( 1,13, 8, 3,15,10, 5,17,12, 7, 2,14, 9, 4,16,11, 6)(18,21,24,27,30,33,19,22, 25,28,31,34,20,23,26,29,32)$
	$ 17, 17 $	$8$	$17$	$( 1, 3, 5, 7, 9,11,13,15,17, 2, 4, 6, 8,10,12,14,16)(18,27,19,28,20,29,21,30, 22,31,23,32,24,33,25,34,26)$
	$ 17, 17 $	$8$	$17$	$( 1, 7,13, 2, 8,14, 3, 9,15, 4,10,16, 5,11,17, 6,12)(18,28,21,31,24,34,27,20, 30,23,33,26,19,29,22,32,25)$
	$ 17, 17 $	$8$	$17$	$( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17)(18,31,27,23,19,32,28,24, 20,33,29,25,21,34,30,26,22)$
	$ 17, 17 $	$8$	$17$	$( 1, 4, 7,10,13,16, 2, 5, 8,11,14,17, 3, 6, 9,12,15)(18,23,28,33,21,26,31,19, 24,29,34,22,27,32,20,25,30)$
	$ 17, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $	$16$	$17$	$(18,21,24,27,30,33,19,22,25,28,31,34,20,23,26,29,32)$
	$ 17, 17 $	$16$	$17$	$( 1, 9,17, 8,16, 7,15, 6,14, 5,13, 4,12, 3,11, 2,10)(18,23,28,33,21,26,31,19, 24,29,34,22,27,32,20,25,30)$
	$ 17, 17 $	$16$	$17$	$( 1, 8,15, 5,12, 2, 9,16, 6,13, 3,10,17, 7,14, 4,11)(18,27,19,28,20,29,21,30, 22,31,23,32,24,33,25,34,26)$
	$ 17, 17 $	$8$	$17$	$( 1,13, 8, 3,15,10, 5,17,12, 7, 2,14, 9, 4,16,11, 6)(18,24,30,19,25,31,20,26, 32,21,27,33,22,28,34,23,29)$
	$ 17, 17 $	$16$	$17$	$( 1, 3, 5, 7, 9,11,13,15,17, 2, 4, 6, 8,10,12,14,16)(18,30,25,20,32,27,22,34, 29,24,19,31,26,21,33,28,23)$
	$ 17, 17 $	$16$	$17$	$( 1, 7,13, 2, 8,14, 3, 9,15, 4,10,16, 5,11,17, 6,12)(18,31,27,23,19,32,28,24, 20,33,29,25,21,34,30,26,22)$
	$ 17, 17 $	$8$	$17$	$( 1,10, 2,11, 3,12, 4,13, 5,14, 6,15, 7,16, 8,17, 9)(18,19,20,21,22,23,24,25, 26,27,28,29,30,31,32,33,34)$
	$ 17, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $	$16$	$17$	$(18,27,19,28,20,29,21,30,22,31,23,32,24,33,25,34,26)$
	$ 17, 17 $	$16$	$17$	$( 1, 9,17, 8,16, 7,15, 6,14, 5,13, 4,12, 3,11, 2,10)(18,29,23,34,28,22,33,27, 21,32,26,20,31,25,19,30,24)$
	$ 17, 17 $	$8$	$17$	$( 1, 3, 5, 7, 9,11,13,15,17, 2, 4, 6, 8,10,12,14,16)(18,19,20,21,22,23,24,25, 26,27,28,29,30,31,32,33,34)$
	$ 17, 17 $	$16$	$17$	$( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17)(18,23,28,33,21,26,31,19, 24,29,34,22,27,32,20,25,30)$
	$ 17, 17 $	$8$	$17$	$( 1,11, 4,14, 7,17,10, 3,13, 6,16, 9, 2,12, 5,15, 8)(18,21,24,27,30,33,19,22, 25,28,31,34,20,23,26,29,32)$
	$ 17, 17 $	$16$	$17$	$( 1, 9,17, 8,16, 7,15, 6,14, 5,13, 4,12, 3,11, 2,10)(18,30,25,20,32,27,22,34, 29,24,19,31,26,21,33,28,23)$
	$ 17, 17 $	$8$	$17$	$( 1, 7,13, 2, 8,14, 3, 9,15, 4,10,16, 5,11,17, 6,12)(18,21,24,27,30,33,19,22, 25,28,31,34,20,23,26,29,32)$
	$ 17, 17 $	$8$	$17$	$( 1,14,10, 6, 2,15,11, 7, 3,16,12, 8, 4,17,13, 9, 5)(18,27,19,28,20,29,21,30, 22,31,23,32,24,33,25,34,26)$
	$ 17, 17 $	$16$	$17$	$( 1, 8,15, 5,12, 2, 9,16, 6,13, 3,10,17, 7,14, 4,11)(18,20,22,24,26,28,30,32, 34,19,21,23,25,27,29,31,33)$
	$ 17, 17 $	$8$	$17$	$( 1, 6,11,16, 4, 9,14, 2, 7,12,17, 5,10,15, 3, 8,13)(18,28,21,31,24,34,27,20, 30,23,33,26,19,29,22,32,25)$
	$ 17, 17 $	$8$	$17$	$( 1,16,14,12,10, 8, 6, 4, 2,17,15,13,11, 9, 7, 5, 3)(18,31,27,23,19,32,28,24, 20,33,29,25,21,34,30,26,22)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $	$289$	$2$	$( 2,17)( 3,16)( 4,15)( 5,14)( 6,13)( 7,12)( 8,11)( 9,10)(19,34)(20,33)(21,32) (22,31)(23,30)(24,29)(25,28)(26,27)$
	$ 34 $	$136$	$34$	$( 1,32, 9,34,17,19, 8,21,16,23, 7,25,15,27, 6,29,14,31, 5,33,13,18, 4,20,12, 22, 3,24,11,26, 2,28,10,30)$
	$ 34 $	$136$	$34$	$( 1,34, 8,23,15,29, 5,18,12,24, 2,30, 9,19,16,25, 6,31,13,20, 3,26,10,32,17, 21, 7,27,14,33, 4,22,11,28)$
	$ 34 $	$136$	$34$	$( 1,21, 6,18,11,32,16,29, 4,26, 9,23,14,20, 2,34, 7,31,12,28,17,25, 5,22,10, 19,15,33, 3,30, 8,27,13,24)$
	$ 34 $	$136$	$34$	$( 1,33,17,20,16,24,15,28,14,32,13,19,12,23,11,27,10,31, 9,18, 8,22, 7,26, 6, 30, 5,34, 4,21, 3,25, 2,29)$
	$ 34 $	$136$	$34$	$( 1,18,16,26,14,34,12,25,10,33, 8,24, 6,32, 4,23, 2,31,17,22,15,30,13,21,11, 29, 9,20, 7,28, 5,19, 3,27)$
	$ 34 $	$136$	$34$	$( 1,25, 4,30, 7,18,10,23,13,28,16,33, 2,21, 5,26, 8,31,11,19,14,24,17,29, 3, 34, 6,22, 9,27,12,32,15,20)$
	$ 34 $	$136$	$34$	$( 1,26,12,33, 6,23,17,30,11,20, 5,27,16,34,10,24, 4,31,15,21, 9,28, 3,18,14, 25, 8,32, 2,22,13,29, 7,19)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$68$	$2$	$( 1,31)( 2,27)( 3,23)( 4,19)( 5,32)( 6,28)( 7,24)( 8,20)( 9,33)(10,29)(11,25) (12,21)(13,34)(14,30)(15,26)(16,22)(17,18)$
	$ 34 $	$136$	$34$	$( 1,23, 5,24, 9,25,13,26,17,27, 4,28, 8,29,12,30,16,31, 3,32, 7,33,11,34,15, 18, 2,19, 6,20,10,21,14,22)$
	$ 4, 4, 4, 4, 4, 4, 4, 4, 1, 1 $	$578$	$4$	$( 2,14,17, 5)( 3,10,16, 9)( 4, 6,15,13)( 7,11,12, 8)(19,22,34,31)(20,26,33,27) (21,30,32,23)(24,25,29,28)$
	$ 8, 8, 8, 8, 1, 1 $	$578$	$8$	$( 2,16, 5,10,17, 3,14, 9)( 4,12,13,11,15, 7, 6, 8)(19,26,31,20,34,27,22,33) (21,25,23,24,32,28,30,29)$
	$ 8, 8, 8, 8, 1, 1 $	$578$	$8$	$( 2, 3, 5, 9,17,16,14,10)( 4, 7,13, 8,15,12, 6,11)(19,27,31,33,34,26,22,20) (21,28,23,29,32,25,30,24)$
	$ 34 $	$136$	$34$	$( 1,32,10,19, 2,23,11,27, 3,31,12,18, 4,22,13,26, 5,30,14,34, 6,21,15,25, 7, 29,16,33, 8,20,17,24, 9,28)$
	$ 34 $	$136$	$34$	$( 1,34,14,19,10,21, 6,23, 2,25,15,27,11,29, 7,31, 3,33,16,18,12,20, 8,22, 4, 24,17,26,13,28, 9,30, 5,32)$
	$ 34 $	$136$	$34$	$( 1,33,12,19, 6,22,17,25,11,28, 5,31,16,34,10,20, 4,23,15,26, 9,29, 3,32,14, 18, 8,21, 2,24,13,27, 7,30)$
	$ 34 $	$136$	$34$	$( 1,18,16,19,14,20,12,21,10,22, 8,23, 6,24, 4,25, 2,26,17,27,15,28,13,29,11, 30, 9,31, 7,32, 5,33, 3,34)$
	$ 34 $	$136$	$34$	$( 1,24,11,19, 4,31,14,26, 7,21,17,33,10,28, 3,23,13,18, 6,30,16,25, 9,20, 2, 32,12,27, 5,22,15,34, 8,29)$
	$ 34 $	$136$	$34$	$( 1,25,13,19, 8,30, 3,24,15,18,10,29, 5,23,17,34,12,28, 7,22, 2,33,14,27, 9, 21, 4,32,16,26,11,20, 6,31)$
	$ 34 $	$136$	$34$	$( 1,28, 2,19, 3,27, 4,18, 5,26, 6,34, 7,25, 8,33, 9,24,10,32,11,23,12,31,13, 22,14,30,15,21,16,29,17,20)$
	$ 34 $	$136$	$34$	$( 1,26,15,19,12,29, 9,22, 6,32, 3,25,17,18,14,28,11,21, 8,31, 5,24, 2,34,16, 27,13,20,10,30, 7,23, 4,33)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$68$	$2$	$( 1,19)( 2,27)( 3,18)( 4,26)( 5,34)( 6,25)( 7,33)( 8,24)( 9,32)(10,23)(11,31) (12,22)(13,30)(14,21)(15,29)(16,20)(17,28)$

magma: ConjugacyClasses(G);

Group invariants

Order:	$4624=2^{4} \cdot 17^{2}$	magma: Order(G);
Cyclic:	no	magma: IsCyclic(G);
Abelian:	no	magma: IsAbelian(G);
Solvable:	yes	magma: IsSolvable(G);
Nilpotency class:	not nilpotent
Label:	4624.t	magma: IdentifyGroup(G);
Character table:	49 x 49 character table

magma: CharacterTable(G);

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Group action invariants

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Subfields

Low degree siblings

Conjugacy classes

Group invariants

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	34T19
Degree	$34$
Order	$4624$
Cyclic	no
Abelian	no
Solvable	yes
Primitive	no
$p$-group	no
Group:	$C_{17}^2:D_8$