LMFDB - Galois group: 34T26

Show commands: Magma

magma: G := TransitiveGroup(34, 26);

Group action invariants

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

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

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

Subfields

Degree 2: $C_2$
Degree 17: None

Low degree siblings

34T25, 34T26
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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $	$16$	$17$	$(18,32,29,26,23,20,34,31,28,25,22,19,33,30,27,24,21)$
	$ 17, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $	$16$	$17$	$(18,26,34,25,33,24,32,23,31,22,30,21,29,20,28,19,27)$
	$ 17, 17 $	$16$	$17$	$( 1, 9,17, 8,16, 7,15, 6,14, 5,13, 4,12, 3,11, 2,10)(18,32,29,26,23,20,34,31, 28,25,22,19,33,30,27,24,21)$
	$ 17, 17 $	$16$	$17$	$( 1, 9,17, 8,16, 7,15, 6,14, 5,13, 4,12, 3,11, 2,10)(18,26,34,25,33,24,32,23, 31,22,30,21,29,20,28,19,27)$
	$ 17, 17 $	$8$	$17$	$( 1, 9,17, 8,16, 7,15, 6,14, 5,13, 4,12, 3,11, 2,10)(18,25,32,22,29,19,26,33, 23,30,20,27,34,24,31,21,28)$
	$ 17, 17 $	$16$	$17$	$( 1, 9,17, 8,16, 7,15, 6,14, 5,13, 4,12, 3,11, 2,10)(18,22,26,30,34,21,25,29, 33,20,24,28,32,19,23,27,31)$
	$ 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 $	$16$	$17$	$( 1, 9,17, 8,16, 7,15, 6,14, 5,13, 4,12, 3,11, 2,10)(18,20,22,24,26,28,30,32, 34,19,21,23,25,27,29,31,33)$
	$ 17, 17 $	$16$	$17$	$( 1, 9,17, 8,16, 7,15, 6,14, 5,13, 4,12, 3,11, 2,10)(18,24,30,19,25,31,20,26, 32,21,27,33,22,28,34,23,29)$
	$ 17, 17 $	$16$	$17$	$( 1, 9,17, 8,16, 7,15, 6,14, 5,13, 4,12, 3,11, 2,10)(18,19,20,21,22,23,24,25, 26,27,28,29,30,31,32,33,34)$
	$ 17, 17 $	$16$	$17$	$( 1, 9,17, 8,16, 7,15, 6,14, 5,13, 4,12, 3,11, 2,10)(18,27,19,28,20,29,21,30, 22,31,23,32,24,33,25,34,26)$
	$ 17, 17 $	$8$	$17$	$( 1, 9,17, 8,16, 7,15, 6,14, 5,13, 4,12, 3,11, 2,10)(18,28,21,31,24,34,27,20, 30,23,33,26,19,29,22,32,25)$
	$ 17, 17 $	$16$	$17$	$( 1, 9,17, 8,16, 7,15, 6,14, 5,13, 4,12, 3,11, 2,10)(18,31,27,23,19,32,28,24, 20,33,29,25,21,34,30,26,22)$
	$ 17, 17 $	$16$	$17$	$( 1, 9,17, 8,16, 7,15, 6,14, 5,13, 4,12, 3,11, 2,10)(18,33,31,29,27,25,23,21, 19,34,32,30,28,26,24,22,20)$
	$ 17, 17 $	$16$	$17$	$( 1, 9,17, 8,16, 7,15, 6,14, 5,13, 4,12, 3,11, 2,10)(18,34,33,32,31,30,29,28, 27,26,25,24,23,22,21,20,19)$
	$ 17, 17 $	$16$	$17$	$( 1, 8,15, 5,12, 2, 9,16, 6,13, 3,10,17, 7,14, 4,11)(18,26,34,25,33,24,32,23, 31,22,30,21,29,20,28,19,27)$
	$ 17, 17 $	$8$	$17$	$( 1, 8,15, 5,12, 2, 9,16, 6,13, 3,10,17, 7,14, 4,11)(18,22,26,30,34,21,25,29, 33,20,24,28,32,19,23,27,31)$
	$ 17, 17 $	$16$	$17$	$( 1, 8,15, 5,12, 2, 9,16, 6,13, 3,10,17, 7,14, 4,11)(18,20,22,24,26,28,30,32, 34,19,21,23,25,27,29,31,33)$
	$ 17, 17 $	$16$	$17$	$( 1, 8,15, 5,12, 2, 9,16, 6,13, 3,10,17, 7,14, 4,11)(18,19,20,21,22,23,24,25, 26,27,28,29,30,31,32,33,34)$
	$ 17, 17 $	$8$	$17$	$( 1, 8,15, 5,12, 2, 9,16, 6,13, 3,10,17, 7,14, 4,11)(18,31,27,23,19,32,28,24, 20,33,29,25,21,34,30,26,22)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $	$289$	$2$	$( 2,17)( 3,16)( 4,15)( 5,14)( 6,13)( 7,12)( 8,11)( 9,10)(19,34)(20,33)(21,32) (22,31)(23,30)(24,29)(25,28)(26,27)$
	$ 4, 4, 4, 4, 4, 4, 4, 4, 1, 1 $	$289$	$4$	$( 2, 5,17,14)( 3, 9,16,10)( 4,13,15, 6)( 7, 8,12,11)(19,22,34,31)(20,26,33,27) (21,30,32,23)(24,25,29,28)$
	$ 4, 4, 4, 4, 4, 4, 4, 4, 1, 1 $	$289$	$4$	$( 2,14,17, 5)( 3,10,16, 9)( 4, 6,15,13)( 7,11,12, 8)(19,31,34,22)(20,27,33,26) (21,23,32,30)(24,28,29,25)$
	$ 8, 8, 8, 8, 1, 1 $	$578$	$8$	$( 2,16, 5,10,17, 3,14, 9)( 4,12,13,11,15, 7, 6, 8)(19,20,22,26,34,33,31,27) (21,24,30,25,32,29,23,28)$
	$ 8, 8, 8, 8, 1, 1 $	$578$	$8$	$( 2,10,14,16,17, 9, 5, 3)( 4,11, 6,12,15, 8,13, 7)(19,26,31,20,34,27,22,33) (21,25,23,24,32,28,30,29)$
	$ 8, 8, 8, 8, 2 $	$578$	$8$	$( 1,34, 5,28, 4,21,17,27)( 2,24, 9,22, 3,31,13,33)( 6,18, 8,32,16,20,14,23) ( 7,25,12,26,15,30,10,29)(11,19)$
	$ 8, 8, 8, 8, 2 $	$578$	$8$	$( 1,31,15,32,10,28,13,27)( 2,25,11,22, 9,34,17,20)( 3,19, 7,29, 8,23, 4,30) ( 5,24,16,26, 6,18,12,33)(14,21)$
	$ 34 $	$136$	$34$	$( 1,33,16,22,14,28,12,34,10,23, 8,29, 6,18, 4,24, 2,30,17,19,15,25,13,31,11, 20, 9,26, 7,32, 5,21, 3,27)$
	$ 34 $	$136$	$34$	$( 1,30,17,33,16,19,15,22,14,25,13,28,12,31,11,34,10,20, 9,23, 8,26, 7,29, 6, 32, 5,18, 4,21, 3,24, 2,27)$
	$ 34 $	$136$	$34$	$( 1,23, 8,19,15,32, 5,28,12,24, 2,20, 9,33,16,29, 6,25,13,21, 3,34,10,30,17, 26, 7,22,14,18, 4,31,11,27)$
	$ 34 $	$136$	$34$	$( 1,18, 4,26, 7,34,10,25,13,33,16,24, 2,32, 5,23, 8,31,11,22,14,30,17,21, 3, 29, 6,20, 9,28,12,19,15,27)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$34$	$2$	$( 1,27)( 2,24)( 3,21)( 4,18)( 5,32)( 6,29)( 7,26)( 8,23)( 9,20)(10,34)(11,31) (12,28)(13,25)(14,22)(15,19)(16,33)(17,30)$
	$ 4, 4, 4, 4, 4, 4, 4, 4, 2 $	$578$	$4$	$( 1,27)( 2,32,17,22)( 3,20,16,34)( 4,25,15,29)( 5,30,14,24)( 6,18,13,19) ( 7,23,12,31)( 8,28,11,26)( 9,33,10,21)$

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.z	magma: IdentifyGroup(G);
Character table:	34 x 34 character table

magma: CharacterTable(G);

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Group action invariants

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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	34T26
Degree	$34$
Order	$4624$
Cyclic	no
Abelian	no
Solvable	yes
Primitive	no
$p$-group	no
Group:	$C_{17}^2:\OD_{16}$