LMFDB - Galois group: 34T33

Show commands: Magma

magma: G := TransitiveGroup(34, 33);

Group invariants

Abstract group:	$C_{17}^2:Q_{32}$	magma: IdentifyGroup(G);
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	magma: NilpotencyClass(G);

Group action invariants

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

Low degree resolvents

$\card{(G/N)}$ Galois groups for stem field(s)
$2$: $C_2$ x 3
$4$: $C_2^2$
$8$: $D_{4}$
$16$: $D_{8}$
$32$: 32T51

Resolvents shown for degrees $\leq 47$

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$ x 3
$4$:	$C_2^2$
$8$:	$D_{4}$
$16$:	$D_{8}$
$32$:	32T51

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	Index	Representative
1A	$1^{34}$	$1$	$1$	$0$	$()$
2A	$2^{16},1^{2}$	$289$	$2$	$16$	$( 1,11)( 2,10)( 3, 9)( 4, 8)( 5, 7)(12,17)(13,16)(14,15)(18,25)(19,24)(20,23)(21,22)(26,34)(27,33)(28,32)(29,31)$
4A	$4^{8},1^{2}$	$578$	$4$	$24$	$( 1, 9,11, 3)( 2, 5,10, 7)( 4,14, 8,15)(12,16,17,13)(18,33,25,27)(19,20,24,23)(21,28,22,32)(26,31,34,29)$
4B	$4^{8},2$	$2312$	$4$	$25$	$( 1,21)( 2,33,17,26)( 3,28,16,31)( 4,23,15,19)( 5,18,14,24)( 6,30,13,29)( 7,25,12,34)( 8,20,11,22)( 9,32,10,27)$
4C	$4^{8},2$	$2312$	$4$	$25$	$( 1,18)( 2,34,17,19)( 3,33,16,20)( 4,32,15,21)( 5,31,14,22)( 6,30,13,23)( 7,29,12,24)( 8,28,11,25)( 9,27,10,26)$
8A1	$8^{4},1^{2}$	$578$	$8$	$28$	$( 1,17, 9,13,11,12, 3,16)( 2, 8, 5,15,10, 4, 7,14)(18,20,33,24,25,23,27,19)(21,31,28,34,22,29,32,26)$
8A3	$8^{4},1^{2}$	$578$	$8$	$28$	$( 1,12, 9,16,11,17, 3,13)( 2, 4, 5,14,10, 8, 7,15)(18,23,33,19,25,20,27,24)(21,29,28,26,22,31,32,34)$
16A1	$16^{2},1^{2}$	$578$	$16$	$30$	$( 1,15,17,10, 9, 4,13, 7,11,14,12, 2, 3, 8,16, 5)(18,31,20,28,33,34,24,22,25,29,23,32,27,26,19,21)$
16A3	$16^{2},1^{2}$	$578$	$16$	$30$	$( 1, 8,12, 7, 9,15,16, 2,11, 4,17, 5, 3,14,13,10)(18,26,23,22,33,31,19,32,25,34,20,21,27,29,24,28)$
16A5	$16^{2},1^{2}$	$578$	$16$	$30$	$( 1, 4,12, 5, 9,14,16,10,11, 8,17, 7, 3,15,13, 2)(18,34,23,21,33,29,19,28,25,26,20,22,27,31,24,32)$
16A7	$16^{2},1^{2}$	$578$	$16$	$30$	$( 1,14,17, 2, 9, 8,13, 5,11,15,12,10, 3, 4,16, 7)(18,29,20,32,33,26,24,21,25,31,23,28,27,34,19,22)$
17A	$17,1^{17}$	$32$	$17$	$16$	$( 1, 3, 5, 7, 9,11,13,15,17, 2, 4, 6, 8,10,12,14,16)$
17B1	$17^{2}$	$32$	$17$	$32$	$( 1,10, 2,11, 3,12, 4,13, 5,14, 6,15, 7,16, 8,17, 9)(18,23,28,33,21,26,31,19,24,29,34,22,27,32,20,25,30)$
17B2	$17^{2}$	$32$	$17$	$32$	$( 1,17,16,15,14,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(18,28,21,31,24,34,27,20,30,23,33,26,19,29,22,32,25)$
17B3	$17^{2}$	$32$	$17$	$32$	$( 1,13, 8, 3,15,10, 5,17,12, 7, 2,14, 9, 4,16,11, 6)(18,31,27,23,19,32,28,24,20,33,29,25,21,34,30,26,22)$
17B6	$17^{2}$	$32$	$17$	$32$	$( 1, 7,13, 2, 8,14, 3, 9,15, 4,10,16, 5,11,17, 6,12)(18,33,31,29,27,25,23,21,19,34,32,30,28,26,24,22,20)$
17C1	$17^{2}$	$32$	$17$	$32$	$( 1, 8,15, 5,12, 2, 9,16, 6,13, 3,10,17, 7,14, 4,11)(18,23,28,33,21,26,31,19,24,29,34,22,27,32,20,25,30)$
17C2	$17^{2}$	$32$	$17$	$32$	$( 1, 9,17, 8,16, 7,15, 6,14, 5,13, 4,12, 3,11, 2,10)(18,31,27,23,19,32,28,24,20,33,29,25,21,34,30,26,22)$
17C3	$17^{2}$	$32$	$17$	$32$	$( 1,15,12, 9, 6, 3,17,14,11, 8, 5, 2,16,13,10, 7, 4)(18,28,21,31,24,34,27,20,30,23,33,26,19,29,22,32,25)$
17C6	$17^{2}$	$32$	$17$	$32$	$( 1, 5, 9,13,17, 4, 8,12,16, 3, 7,11,15, 2, 6,10,14)(18,33,31,29,27,25,23,21,19,34,32,30,28,26,24,22,20)$