LMFDB - Galois group: 34T35

Show commands: Magma

magma: G := TransitiveGroup(34, 35);

Group action invariants

Degree $n$:	$34$	magma: t, n := TransitiveGroupIdentification(G); n;
Transitive number $t$:	$35$	magma: t, n := TransitiveGroupIdentification(G); t;
Group:	$D_{17}^2.(C_2\times C_4)$
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:	(1,14,16,15,7,11,9,10)(2,5,12,17,6,3,13,8)(18,21,28,33,22,19,29,24)(23,27,25,26,34,30,32,31), (1,5,14,13,15,11,2,3)(4,16,9,6,12,17,7,10)(18,22,30,29,27,23,32,33)(19,24,34,20,26,21,28,25), (1,25,17,19,13,29,14,18)(2,31,4,26,12,23,10,28)(3,20,8,33,11,34,6,21)(5,32,16,30,9,22,15,24)(7,27)	magma: Generators(G);

Low degree resolvents

|G/N| Galois groups for stem field(s)
$2$: $C_2$ x 7
$4$: $C_4$ x 4, $C_2^2$ x 7
$8$: $C_4\times C_2$ x 6, $C_2^3$
$16$: $C_4\times C_2^2$
$32$: $(C_8:C_2):C_2$

Resolvents shown for degrees $\leq 47$

\|G/N\|	Galois groups for stem field(s)
$2$:	$C_2$ x 7
$4$:	$C_4$ x 4, $C_2^2$ x 7
$8$:	$C_4\times C_2$ x 6, $C_2^3$
$16$:	$C_4\times C_2^2$
$32$:	$(C_8:C_2):C_2$

Subfields

Degree 2: $C_2$
Degree 17: None

Low degree siblings

34T35 x 2
Siblings are shown 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 $	$32$	$17$	$( 1, 4, 7,10,13,16, 2, 5, 8,11,14,17, 3, 6, 9,12,15)(18,28,21,31,24,34,27,20, 30,23,33,26,19,29,22,32,25)$
	$ 17, 17 $	$32$	$17$	$( 1,10, 2,11, 3,12, 4,13, 5,14, 6,15, 7,16, 8,17, 9)(18,31,27,23,19,32,28,24, 20,33,29,25,21,34,30,26,22)$
	$ 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 $	$16$	$17$	$( 1, 4, 7,10,13,16, 2, 5, 8,11,14,17, 3, 6, 9,12,15)(18,20,22,24,26,28,30,32, 34,19,21,23,25,27,29,31,33)$
	$ 17, 17 $	$32$	$17$	$( 1,10, 2,11, 3,12, 4,13, 5,14, 6,15, 7,16, 8,17, 9)(18,23,28,33,21,26,31,19, 24,29,34,22,27,32,20,25,30)$
	$ 17, 17 $	$32$	$17$	$( 1,11, 4,14, 7,17,10, 3,13, 6,16, 9, 2,12, 5,15, 8)(18,32,29,26,23,20,34,31, 28,25,22,19,33,30,27,24,21)$
	$ 17, 17 $	$32$	$17$	$( 1, 6,11,16, 4, 9,14, 2, 7,12,17, 5,10,15, 3, 8,13)(18,21,24,27,30,33,19,22, 25,28,31,34,20,23,26,29,32)$
	$ 17, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $	$16$	$17$	$( 1,17,16,15,14,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)$
	$ 17, 17 $	$32$	$17$	$( 1,15,12, 9, 6, 3,17,14,11, 8, 5, 2,16,13,10, 7, 4)(18,34,33,32,31,30,29,28, 27,26,25,24,23,22,21,20,19)$
	$ 17, 17 $	$16$	$17$	$( 1, 8,15, 5,12, 2, 9,16, 6,13, 3,10,17, 7,14, 4,11)(18,22,26,30,34,21,25,29, 33,20,24,28,32,19,23,27,31)$
	$ 17, 17 $	$16$	$17$	$( 1, 3, 5, 7, 9,11,13,15,17, 2, 4, 6, 8,10,12,14,16)(18,28,21,31,24,34,27,20, 30,23,33,26,19,29,22,32,25)$
	$ 17, 17 $	$16$	$17$	$( 1, 5, 9,13,17, 4, 8,12,16, 3, 7,11,15, 2, 6,10,14)(18,30,25,20,32,27,22,34, 29,24,19,31,26,21,33,28,23)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $	$289$	$2$	$( 2,17)( 3,16)( 4,15)( 5,14)( 6,13)( 7,12)( 8,11)( 9,10)(19,34)(20,33)(21,32) (22,31)(23,30)(24,29)(25,28)(26,27)$
	$ 4, 4, 4, 4, 4, 4, 4, 4, 1, 1 $	$289$	$4$	$( 2,14,17, 5)( 3,10,16, 9)( 4, 6,15,13)( 7,11,12, 8)(19,31,34,22)(20,27,33,26) (21,23,32,30)(24,28,29,25)$
	$ 4, 4, 4, 4, 4, 4, 4, 4, 1, 1 $	$289$	$4$	$( 2, 5,17,14)( 3, 9,16,10)( 4,13,15, 6)( 7, 8,12,11)(19,22,34,31)(20,26,33,27) (21,30,32,23)(24,25,29,28)$
	$ 8, 8, 8, 8, 1, 1 $	$289$	$8$	$( 2, 9,14, 3,17,10, 5,16)( 4, 8, 6, 7,15,11,13,12)(19,26,31,20,34,27,22,33) (21,25,23,24,32,28,30,29)$
	$ 8, 8, 8, 8, 1, 1 $	$289$	$8$	$( 2,10,14,16,17, 9, 5, 3)( 4,11, 6,12,15, 8,13, 7)(19,27,31,33,34,26,22,20) (21,28,23,29,32,25,30,24)$
	$ 8, 8, 8, 8, 1, 1 $	$289$	$8$	$( 2, 3, 5, 9,17,16,14,10)( 4, 7,13, 8,15,12, 6,11)(19,20,22,26,34,33,31,27) (21,24,30,25,32,29,23,28)$
	$ 8, 8, 8, 8, 1, 1 $	$289$	$8$	$( 2,16, 5,10,17, 3,14, 9)( 4,12,13,11,15, 7, 6, 8)(19,33,22,27,34,20,31,26) (21,29,30,28,32,24,23,25)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $	$34$	$2$	$(19,34)(20,33)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)$
	$ 17, 2, 2, 2, 2, 2, 2, 2, 2, 1 $	$272$	$34$	$( 1, 4, 7,10,13,16, 2, 5, 8,11,14,17, 3, 6, 9,12,15)(18,28)(19,27)(20,26) (21,25)(22,24)(29,34)(30,33)(31,32)$
	$ 17, 2, 2, 2, 2, 2, 2, 2, 2, 1 $	$272$	$34$	$( 1,10, 2,11, 3,12, 4,13, 5,14, 6,15, 7,16, 8,17, 9)(18,31)(19,30)(20,29) (21,28)(22,27)(23,26)(24,25)(32,34)$
	$ 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 $	$578$	$8$	$( 2, 9,14, 3,17,10, 5,16)( 4, 8, 6, 7,15,11,13,12)(19,27,31,33,34,26,22,20) (21,28,23,29,32,25,30,24)$
	$ 8, 8, 8, 8, 1, 1 $	$578$	$8$	$( 2, 3, 5, 9,17,16,14,10)( 4, 7,13, 8,15,12, 6,11)(19,33,22,27,34,20,31,26) (21,29,30,28,32,24,23,25)$
	$ 8, 8, 8, 8, 2 $	$578$	$8$	$( 1,25,17,19,13,29,14,18)( 2,31, 4,26,12,23,10,28)( 3,20, 8,33,11,34, 6,21) ( 5,32,16,30, 9,22,15,24)( 7,27)$
	$ 8, 8, 8, 8, 2 $	$578$	$8$	$( 1,24, 2,34,15,28,14,18)( 3,27,11,22,13,25, 5,30)( 4,20, 7,33,12,32, 9,19) ( 6,23,16,21,10,29,17,31)( 8,26)$
	$ 34 $	$272$	$34$	$( 1,23, 5,28, 9,33,13,21,17,26, 4,31, 8,19,12,24,16,29, 3,34, 7,22,11,27,15, 32, 2,20, 6,25,10,30,14,18)$
	$ 34 $	$272$	$34$	$( 1,19, 4,27, 7,18,10,26,13,34,16,25, 2,33, 5,24, 8,32,11,23,14,31,17,22, 3, 30, 6,21, 9,29,12,20,15,28)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$34$	$2$	$( 1,24)( 2,21)( 3,18)( 4,32)( 5,29)( 6,26)( 7,23)( 8,20)( 9,34)(10,31)(11,28) (12,25)(13,22)(14,19)(15,33)(16,30)(17,27)$
	$ 4, 4, 4, 4, 4, 4, 4, 4, 2 $	$578$	$4$	$( 1,32,14,18)( 2,27,13,23)( 3,22,12,28)( 4,34,11,33)( 5,29,10,21)( 6,24, 9,26) ( 7,19, 8,31)(15,30,17,20)(16,25)$
	$ 8, 8, 8, 8, 2 $	$578$	$8$	$( 1,28, 2,22,15,29,14,18)( 3,33,11,19,13,24, 5,21)( 4,27, 7,26,12,30, 9,31) ( 6,32,16,23,10,25,17,34)( 8,20)$
	$ 8, 8, 8, 8, 2 $	$578$	$8$	$( 1,29,17,22,13,28,14,18)( 2,19, 4,33,12,21,10,24)( 3,26, 8,27,11,31, 6,30) ( 5,23,16,32, 9,34,15,25)( 7,20)$
	$ 4, 4, 4, 4, 4, 4, 4, 4, 2 $	$578$	$4$	$( 1,30,14,18)( 2,33,13,32)( 3,19,12,29)( 4,22,11,26)( 5,25,10,23)( 6,28, 9,20) ( 7,31, 8,34)(15,21,17,27)(16,24)$
	$ 34 $	$272$	$34$	$( 1,21, 5,24, 9,27,13,30,17,33, 4,19, 8,22,12,25,16,28, 3,31, 7,34,11,20,15, 23, 2,26, 6,29,10,32,14,18)$
	$ 34 $	$272$	$34$	$( 1,19, 4,34, 7,32,10,30,13,28,16,26, 2,24, 5,22, 8,20,11,18,14,33,17,31, 3, 29, 6,27, 9,25,12,23,15,21)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$34$	$2$	$( 1,30)( 2,18)( 3,23)( 4,28)( 5,33)( 6,21)( 7,26)( 8,31)( 9,19)(10,24)(11,29) (12,34)(13,22)(14,27)(15,32)(16,20)(17,25)$

magma: ConjugacyClasses(G);

Group invariants

Order:	$9248=2^{5} \cdot 17^{2}$	magma: Order(G);
Cyclic:	no	magma: IsCyclic(G);
Abelian:	no	magma: IsAbelian(G);
Solvable:	yes	magma: IsSolvable(G);
Nilpotency class:	not nilpotent
Label:	9248.v	magma: IdentifyGroup(G);
Character table:	38 x 38 character table

magma: CharacterTable(G);

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Group action invariants

Low degree resolvents

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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	34T35
Degree	$34$
Order	$9248$
Cyclic	no
Abelian	no
Solvable	yes
Primitive	no
$p$-group	no
Group:	$D_{17}^2.(C_2\times C_4)$