Show commands:
Magma
magma: G := TransitiveGroup(34, 2);
Group invariants
Abstract group: | $D_{17}$ | magma: IdentifyGroup(G);
| |
Order: | $34=2 \cdot 17$ | 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 $t$: | $2$ | magma: t, n := TransitiveGroupIdentification(G); t;
| |
Parity: | $-1$ | magma: IsEven(G);
| |
Primitive: | no | magma: IsPrimitive(G);
| |
$\card{\Aut(F/K)}$: | $34$ | magma: Order(Centralizer(SymmetricGroup(n), G));
| |
Generators: | $(1,32)(2,31)(3,29)(4,30)(5,27)(6,28)(7,26)(8,25)(9,23)(10,24)(11,22)(12,21)(13,19)(14,20)(15,17)(16,18)(33,34)$, $(1,9,18,26,34,8,15,24,31,6,14,22,29,4,12,19,27)(2,10,17,25,33,7,16,23,32,5,13,21,30,3,11,20,28)$ | magma: Generators(G);
|
Low degree resolvents
$\card{(G/N)}$ Galois groups for stem field(s) $2$: $C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 17: $D_{17}$
Low degree siblings
17T2Siblings 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 | Index | Representative |
1A | $1^{34}$ | $1$ | $1$ | $0$ | $()$ |
2A | $2^{17}$ | $17$ | $2$ | $17$ | $( 1, 3)( 2, 4)( 5,34)( 6,33)( 7,31)( 8,32)( 9,30)(10,29)(11,27)(12,28)(13,26)(14,25)(15,23)(16,24)(17,22)(18,21)(19,20)$ |
17A1 | $17^{2}$ | $2$ | $17$ | $32$ | $( 1, 4, 6, 8, 9,12,14,15,18,19,22,24,26,27,29,31,34)( 2, 3, 5, 7,10,11,13,16,17,20,21,23,25,28,30,32,33)$ |
17A2 | $17^{2}$ | $2$ | $17$ | $32$ | $( 1, 6, 9,14,18,22,26,29,34, 4, 8,12,15,19,24,27,31)( 2, 5,10,13,17,21,25,30,33, 3, 7,11,16,20,23,28,32)$ |
17A3 | $17^{2}$ | $2$ | $17$ | $32$ | $( 1, 8,14,19,26,31, 4, 9,15,22,27,34, 6,12,18,24,29)( 2, 7,13,20,25,32, 3,10,16,21,28,33, 5,11,17,23,30)$ |
17A4 | $17^{2}$ | $2$ | $17$ | $32$ | $( 1, 9,18,26,34, 8,15,24,31, 6,14,22,29, 4,12,19,27)( 2,10,17,25,33, 7,16,23,32, 5,13,21,30, 3,11,20,28)$ |
17A5 | $17^{2}$ | $2$ | $17$ | $32$ | $( 1,12,22,31, 8,18,27, 4,14,24,34, 9,19,29, 6,15,26)( 2,11,21,32, 7,17,28, 3,13,23,33,10,20,30, 5,16,25)$ |
17A6 | $17^{2}$ | $2$ | $17$ | $32$ | $( 1,14,26, 4,15,27, 6,18,29, 8,19,31, 9,22,34,12,24)( 2,13,25, 3,16,28, 5,17,30, 7,20,32,10,21,33,11,23)$ |
17A7 | $17^{2}$ | $2$ | $17$ | $32$ | $( 1,15,29, 9,24, 4,18,31,12,26, 6,19,34,14,27, 8,22)( 2,16,30,10,23, 3,17,32,11,25, 5,20,33,13,28, 7,21)$ |
17A8 | $17^{2}$ | $2$ | $17$ | $32$ | $( 1,18,34,15,31,14,29,12,27, 9,26, 8,24, 6,22, 4,19)( 2,17,33,16,32,13,30,11,28,10,25, 7,23, 5,21, 3,20)$ |
Malle's constant $a(G)$: $1/17$
magma: ConjugacyClasses(G);
Character table
1A | 2A | 17A1 | 17A2 | 17A3 | 17A4 | 17A5 | 17A6 | 17A7 | 17A8 | ||
Size | 1 | 17 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | |
2 P | 1A | 1A | 17A2 | 17A4 | 17A6 | 17A8 | 17A7 | 17A5 | 17A3 | 17A1 | |
17 P | 1A | 2A | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 1A | |
Type | |||||||||||
34.1.1a | R | ||||||||||
34.1.1b | R | ||||||||||
34.1.2a1 | R | ||||||||||
34.1.2a2 | R | ||||||||||
34.1.2a3 | R | ||||||||||
34.1.2a4 | R | ||||||||||
34.1.2a5 | R | ||||||||||
34.1.2a6 | R | ||||||||||
34.1.2a7 | R | ||||||||||
34.1.2a8 | R |
magma: CharacterTable(G);
Regular extensions
Data not computed