LMFDB - Galois group: 34T14

Show commands: Magma

magma: G := TransitiveGroup(34, 14);

Group action invariants

Degree $n$:	$34$	magma: t, n := TransitiveGroupIdentification(G); n;
Transitive number $t$:	$14$	magma: t, n := TransitiveGroupIdentification(G); t;
Group:	$C_{17}^2:Q_8$
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:	(1,25,16,18)(2,20,15,23)(3,32,14,28)(4,27,13,33)(5,22,12,21)(6,34,11,26)(7,29,10,31)(8,24,9,19)(17,30), (1,17,13,14)(2,4,12,10)(3,8,11,6)(5,16,9,15)(18,20,29,27)(19,33,28,31)(21,25,26,22)(23,34,24,30)	magma: Generators(G);

Low degree resolvents

|G/N| Galois groups for stem field(s)
$2$: $C_2$ x 3
$4$: $C_2^2$
$8$: $Q_8$

Resolvents shown for degrees $\leq 47$

\|G/N\|	Galois groups for stem field(s)
$2$:	$C_2$ x 3
$4$:	$C_2^2$
$8$:	$Q_8$

Subfields

Degree 2: $C_2$
Degree 17: None

Low degree siblings

34T14 x 2
Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

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

magma: ConjugacyClasses(G);

Group invariants

Order:	$2312=2^{3} \cdot 17^{2}$	magma: Order(G);
Cyclic:	no	magma: IsCyclic(G);
Abelian:	no	magma: IsAbelian(G);
Solvable:	yes	magma: IsSolvable(G);
Nilpotency class:	not nilpotent
Label:	2312.n	magma: IdentifyGroup(G);

Character table: not available.

magma: CharacterTable(G);

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Group action invariants

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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	34T14
Degree	$34$
Order	$2312$
Cyclic	no
Abelian	no
Solvable	yes
Primitive	no
$p$-group	no
Group:	$C_{17}^2:Q_8$