Show commands:
Magma
magma: G := TransitiveGroup(34, 30);
Group invariants
| Abstract group: | $C_2^8.D_{17}$ | magma: IdentifyGroup(G);
| |
| Order: | $8704=2^{9} \cdot 17$ | 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 $t$: | $30$ | magma: t, n := TransitiveGroupIdentification(G); t;
| |
| Parity: | $1$ | magma: IsEven(G);
| |
| Primitive: | no | magma: IsPrimitive(G);
| |
| $\card{\Aut(F/K)}$: | $2$ | magma: Order(Centralizer(SymmetricGroup(n), G));
| |
| Generators: | $(1,19)(2,20)(3,18)(4,17)(5,16)(6,15)(7,13,8,14)(9,11,10,12)(21,33,22,34)(23,32)(24,31)(25,29)(26,30)(27,28)$, $(1,13,2,14)(3,12)(4,11)(5,9,6,10)(15,34,16,33)(17,31,18,32)(19,29,20,30)(21,28,22,27)(23,25)(24,26)$ | magma: Generators(G);
|
Low degree resolvents
$\card{(G/N)}$ Galois groups for stem field(s) $2$: $C_2$ $34$: $D_{17}$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 17: $D_{17}$
Low degree siblings
34T30 x 14, 34T31 x 15Siblings 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 | $2^{6},1^{22}$ | $17$ | $2$ | $6$ | $( 1, 2)(17,18)(19,20)(23,24)(29,30)(33,34)$ |
| 2B | $2^{10},1^{14}$ | $17$ | $2$ | $10$ | $( 1, 2)( 3, 4)( 5, 6)(11,12)(15,16)(21,22)(23,24)(25,26)(29,30)(31,32)$ |
| 2C | $2^{6},1^{22}$ | $17$ | $2$ | $6$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)(15,16)(27,28)$ |
| 2D | $2^{6},1^{22}$ | $17$ | $2$ | $6$ | $( 5, 6)(11,12)(13,14)(23,24)(25,26)(31,32)$ |
| 2E | $2^{8},1^{18}$ | $17$ | $2$ | $8$ | $( 1, 2)( 3, 4)( 5, 6)( 9,10)(23,24)(27,28)(29,30)(31,32)$ |
| 2F | $2^{10},1^{14}$ | $17$ | $2$ | $10$ | $( 3, 4)( 5, 6)(11,12)(15,16)(17,18)(19,20)(21,22)(25,26)(31,32)(33,34)$ |
| 2G | $2^{8},1^{18}$ | $17$ | $2$ | $8$ | $( 1, 2)( 3, 4)( 9,10)(11,12)(13,14)(25,26)(27,28)(29,30)$ |
| 2H | $2^{10},1^{14}$ | $17$ | $2$ | $10$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)(13,14)(19,20)(25,26)(27,28)(29,30)(31,32)$ |
| 2I | $2^{8},1^{18}$ | $17$ | $2$ | $8$ | $( 3, 4)( 5, 6)( 9,10)(17,18)(19,20)(27,28)(31,32)(33,34)$ |
| 2J | $2^{12},1^{10}$ | $17$ | $2$ | $12$ | $( 3, 4)( 7, 8)(11,12)(13,14)(15,16)(17,18)(19,20)(25,26)(27,28)(29,30)(31,32)(33,34)$ |
| 2K | $2^{6},1^{22}$ | $17$ | $2$ | $6$ | $( 5, 6)(11,12)(15,16)(19,20)(23,24)(29,30)$ |
| 2L | $2^{10},1^{14}$ | $17$ | $2$ | $10$ | $( 3, 4)( 5, 6)( 7, 8)(15,16)(17,18)(19,20)(23,24)(27,28)(29,30)(33,34)$ |
| 2M | $2^{8},1^{18}$ | $17$ | $2$ | $8$ | $( 5, 6)( 9,10)(13,14)(15,16)(21,22)(23,24)(27,28)(31,32)$ |
| 2N | $2^{12},1^{10}$ | $17$ | $2$ | $12$ | $( 1, 2)( 9,10)(11,12)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(33,34)$ |
| 2O | $2^{8},1^{18}$ | $17$ | $2$ | $8$ | $( 7, 8)(11,12)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)$ |
| 2P | $2^{16},1^{2}$ | $272$ | $2$ | $16$ | $( 1,13)( 2,14)( 3,12)( 4,11)( 5, 9)( 6,10)(15,34)(16,33)(17,31)(18,32)(19,29)(20,30)(21,28)(22,27)(23,26)(24,25)$ |
| 4A | $4^{4},2^{8},1^{2}$ | $272$ | $4$ | $20$ | $( 1,13)( 2,14)( 3,12)( 4,11)( 5,10, 6, 9)(15,34)(16,33)(17,32)(18,31)(19,29,20,30)(21,28,22,27)(23,26,24,25)$ |
| 4B | $4^{3},2^{11}$ | $272$ | $4$ | $20$ | $( 1,13, 2,14)( 3,12)( 4,11)( 5, 9)( 6,10)( 7, 8)(15,34)(16,33)(17,32)(18,31)(19,29)(20,30)(21,27,22,28)(23,26,24,25)$ |
| 4C | $4^{5},2^{7}$ | $272$ | $4$ | $22$ | $( 1,13, 2,14)( 3,11, 4,12)( 5,10, 6, 9)( 7, 8)(15,34,16,33)(17,32)(18,31)(19,29)(20,30)(21,28,22,27)(23,25)(24,26)$ |
| 4D | $4^{4},2^{8},1^{2}$ | $272$ | $4$ | $20$ | $( 1,13)( 2,14)( 3,11, 4,12)( 5,10, 6, 9)(15,34,16,33)(17,31)(18,32)(19,29)(20,30)(21,27)(22,28)(23,25,24,26)$ |
| 4E | $4^{5},2^{7}$ | $272$ | $4$ | $22$ | $( 1,13)( 2,14)( 3,11, 4,12)( 5,10, 6, 9)( 7, 8)(15,33)(16,34)(17,32,18,31)(19,30)(20,29)(21,28,22,27)(23,25,24,26)$ |
| 4F | $4^{4},2^{8},1^{2}$ | $272$ | $4$ | $20$ | $( 1,13, 2,14)( 3,11, 4,12)( 5,10, 6, 9)(15,33)(16,34)(17,31,18,32)(19,30)(20,29)(21,27)(22,28)(23,25)(24,26)$ |
| 4G | $4^{6},2^{4},1^{2}$ | $272$ | $4$ | $22$ | $( 1,13, 2,14)( 3,11, 4,12)( 5, 9)( 6,10)(15,33)(16,34)(17,32,18,31)(19,30,20,29)(21,27,22,28)(23,25,24,26)$ |
| 4H | $4^{4},2^{8},1^{2}$ | $272$ | $4$ | $20$ | $( 1,13)( 2,14)( 3,11, 4,12)( 5, 9)( 6,10)(15,34,16,33)(17,32)(18,31)(19,29,20,30)(21,27,22,28)(23,25)(24,26)$ |
| 4I | $4^{3},2^{11}$ | $272$ | $4$ | $20$ | $( 1,13)( 2,14)( 3,11, 4,12)( 5, 9)( 6,10)( 7, 8)(15,33)(16,34)(17,31,18,32)(19,30,20,29)(21,28)(22,27)(23,25)(24,26)$ |
| 4J | $4^{4},2^{8},1^{2}$ | $272$ | $4$ | $20$ | $( 1,13, 2,14)( 3,12)( 4,11)( 5, 9)( 6,10)(15,33,16,34)(17,31,18,32)(19,30)(20,29)(21,28)(22,27)(23,26,24,25)$ |
| 4K | $4^{3},2^{11}$ | $272$ | $4$ | $20$ | $( 1,13, 2,14)( 3,12)( 4,11)( 5,10, 6, 9)( 7, 8)(15,34)(16,33)(17,31)(18,32)(19,29,20,30)(21,27)(22,28)(23,26)(24,25)$ |
| 4L | $4^{6},2^{4},1^{2}$ | $272$ | $4$ | $22$ | $( 1,13, 2,14)( 3,12)( 4,11)( 5,10, 6, 9)(15,33,16,34)(17,32,18,31)(19,30,20,29)(21,28,22,27)(23,26)(24,25)$ |
| 4M | $4^{5},2^{7}$ | $272$ | $4$ | $22$ | $( 1,13, 2,14)( 3,11, 4,12)( 5, 9)( 6,10)( 7, 8)(15,34,16,33)(17,31)(18,32)(19,29,20,30)(21,28)(22,27)(23,25,24,26)$ |
| 4N | $4^{5},2^{7}$ | $272$ | $4$ | $22$ | $( 1,13)( 2,14)( 3,12)( 4,11)( 5,10, 6, 9)( 7, 8)(15,33,16,34)(17,31,18,32)(19,30,20,29)(21,27)(22,28)(23,26,24,25)$ |
| 4O | $4^{3},2^{11}$ | $272$ | $4$ | $20$ | $( 1,13)( 2,14)( 3,12)( 4,11)( 5, 9)( 6,10)( 7, 8)(15,33,16,34)(17,32,18,31)(19,30)(20,29)(21,27,22,28)(23,26)(24,25)$ |
| 17A1 | $17^{2}$ | $512$ | $17$ | $32$ | $( 1,24,11,33,22, 9,32,20, 8,29,17, 5,28,16, 3,26,14)( 2,23,12,34,21,10,31,19, 7,30,18, 6,27,15, 4,25,13)$ |
| 17A2 | $17^{2}$ | $512$ | $17$ | $32$ | $( 1,11,22,32, 8,17,28, 3,14,24,33, 9,20,29, 5,16,26)( 2,12,21,31, 7,18,27, 4,13,23,34,10,19,30, 6,15,25)$ |
| 17A3 | $17^{2}$ | $512$ | $17$ | $32$ | $( 1,33,32,29,28,26,24,22,20,17,16,14,11, 9, 8, 5, 3)( 2,34,31,30,27,25,23,21,19,18,15,13,12,10, 7, 6, 4)$ |
| 17A4 | $17^{2}$ | $512$ | $17$ | $32$ | $( 1,22, 8,28,14,33,20, 5,26,11,32,17, 3,24, 9,29,16)( 2,21, 7,27,13,34,19, 6,25,12,31,18, 4,23,10,30,15)$ |
| 17A5 | $17^{2}$ | $512$ | $17$ | $32$ | $( 1, 9,17,26,33, 8,16,24,32, 5,14,22,29, 3,11,20,28)( 2,10,18,25,34, 7,15,23,31, 6,13,21,30, 4,12,19,27)$ |
| 17A6 | $17^{2}$ | $512$ | $17$ | $32$ | $( 1,32,28,24,20,16,11, 8, 3,33,29,26,22,17,14, 9, 5)( 2,31,27,23,19,15,12, 7, 4,34,30,25,21,18,13,10, 6)$ |
| 17A7 | $17^{2}$ | $512$ | $17$ | $32$ | $( 1,20, 3,22, 5,24, 8,26, 9,28,11,29,14,32,16,33,17)( 2,19, 4,21, 6,23, 7,25,10,27,12,30,13,31,15,34,18)$ |
| 17A8 | $17^{2}$ | $512$ | $17$ | $32$ | $( 1, 8,14,20,26,32, 3, 9,16,22,28,33, 5,11,17,24,29)( 2, 7,13,19,25,31, 4,10,15,21,27,34, 6,12,18,23,30)$ |
Malle's constant $a(G)$: $1/6$
magma: ConjugacyClasses(G);
Character table
40 x 40 character tablemagma: CharacterTable(G);
Regular extensions
Data not computed