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Magma
magma: G := TransitiveGroup(34, 31);
Group invariants
| Abstract group: | $C_2^8.D_{17}$ | magma: IdentifyGroup(G);
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| Order: | $8704=2^{9} \cdot 17$ | magma: Order(G);
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| Cyclic: | no | magma: IsCyclic(G);
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| Abelian: | no | magma: IsAbelian(G);
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| Solvable: | yes | magma: IsSolvable(G);
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| Nilpotency class: | not nilpotent | magma: NilpotencyClass(G);
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Group action invariants
| Degree $n$: | $34$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $31$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Parity: | $-1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
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| $\card{\Aut(F/K)}$: | $2$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | $(1,28)(2,27)(3,25)(4,26)(5,23)(6,24)(7,22)(8,21)(9,20)(10,19)(11,17)(12,18)(13,15)(14,16)(29,34)(30,33)(31,32)$, $(1,30,24,18,12,6,34,28,22,15,9,4,32,25,20,14,8)(2,29,23,17,11,5,33,27,21,16,10,3,31,26,19,13,7)$ | magma: Generators(G);
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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 15, 34T31 x 14Siblings 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 | $2^{6},1^{22}$ | $17$ | $2$ | $6$ | $( 7, 8)( 9,10)(13,14)(19,20)(23,24)(25,26)$ |
| 2B | $2^{10},1^{14}$ | $17$ | $2$ | $10$ | $( 3, 4)( 5, 6)( 7, 8)(11,12)(13,14)(17,18)(19,20)(21,22)(27,28)(31,32)$ |
| 2C | $2^{6},1^{22}$ | $17$ | $2$ | $6$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)(15,16)(27,28)$ |
| 2D | $2^{6},1^{22}$ | $17$ | $2$ | $6$ | $( 3, 4)(11,12)(17,18)(19,20)(29,30)(31,32)$ |
| 2E | $2^{8},1^{18}$ | $17$ | $2$ | $8$ | $( 3, 4)( 5, 6)( 7, 8)(11,12)(13,14)(15,16)(19,20)(33,34)$ |
| 2F | $2^{10},1^{14}$ | $17$ | $2$ | $10$ | $( 3, 4)( 5, 6)( 9,10)(11,12)(17,18)(21,22)(23,24)(25,26)(27,28)(31,32)$ |
| 2G | $2^{8},1^{18}$ | $17$ | $2$ | $8$ | $( 5, 6)( 7, 8)(13,14)(15,16)(17,18)(29,30)(31,32)(33,34)$ |
| 2H | $2^{10},1^{14}$ | $17$ | $2$ | $10$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)(11,12)(13,14)(15,16)(17,18)(23,24)(29,30)$ |
| 2I | $2^{8},1^{18}$ | $17$ | $2$ | $8$ | $( 3, 4)( 5, 6)( 9,10)(11,12)(15,16)(23,24)(25,26)(33,34)$ |
| 2J | $2^{12},1^{10}$ | $17$ | $2$ | $12$ | $( 1, 2)( 5, 6)( 9,10)(11,12)(13,14)(15,16)(17,18)(23,24)(25,26)(27,28)(29,30)(31,32)$ |
| 2K | $2^{6},1^{22}$ | $17$ | $2$ | $6$ | $( 3, 4)(13,14)(19,20)(23,24)(27,28)(31,32)$ |
| 2L | $2^{10},1^{14}$ | $17$ | $2$ | $10$ | $( 1, 2)( 3, 4)( 5, 6)( 9,10)(13,14)(15,16)(19,20)(23,24)(25,26)(27,28)$ |
| 2M | $2^{8},1^{18}$ | $17$ | $2$ | $8$ | $( 3, 4)(11,12)(15,16)(19,20)(21,22)(27,28)(29,30)(33,34)$ |
| 2N | $2^{12},1^{10}$ | $17$ | $2$ | $12$ | $( 7, 8)( 9,10)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(31,32)(33,34)$ |
| 2O | $2^{8},1^{18}$ | $17$ | $2$ | $8$ | $( 1, 2)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(31,32)$ |
| 2P | $2^{17}$ | $272$ | $2$ | $17$ | $( 1, 2)( 3,34)( 4,33)( 5,32)( 6,31)( 7,29)( 8,30)( 9,28)(10,27)(11,26)(12,25)(13,24)(14,23)(15,21)(16,22)(17,19)(18,20)$ |
| 4A | $4^{4},2^{9}$ | $272$ | $4$ | $21$ | $( 1, 2)( 3,33, 4,34)( 5,32)( 6,31)( 7,29)( 8,30)( 9,28)(10,27)(11,25)(12,26)(13,23,14,24)(15,22,16,21)(17,20,18,19)$ |
| 4B | $4^{3},2^{10},1^{2}$ | $272$ | $4$ | $19$ | $( 3,34)( 4,33)( 5,32)( 6,31)( 7,29, 8,30)( 9,28)(10,27)(11,25)(12,26)(13,24)(14,23)(15,21,16,22)(17,20,18,19)$ |
| 4C | $4^{5},2^{6},1^{2}$ | $272$ | $4$ | $21$ | $( 3,33, 4,34)( 5,31, 6,32)( 7,29, 8,30)( 9,27,10,28)(11,25)(12,26)(13,24)(14,23)(15,22,16,21)(17,20)(18,19)$ |
| 4D | $4^{4},2^{9}$ | $272$ | $4$ | $21$ | $( 1, 2)( 3,33, 4,34)( 5,31, 6,32)( 7,29)( 8,30)( 9,27,10,28)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,19,18,20)$ |
| 4E | $4^{5},2^{6},1^{2}$ | $272$ | $4$ | $21$ | $( 3,33, 4,34)( 5,31, 6,32)( 7,29)( 8,30)( 9,27)(10,28)(11,26,12,25)(13,23)(14,24)(15,22,16,21)(17,19,18,20)$ |
| 4F | $4^{4},2^{9}$ | $272$ | $4$ | $21$ | $( 1, 2)( 3,33, 4,34)( 5,31, 6,32)( 7,29, 8,30)( 9,27)(10,28)(11,25,12,26)(13,23)(14,24)(15,22)(16,21)(17,20)(18,19)$ |
| 4G | $4^{6},2^{5}$ | $272$ | $4$ | $23$ | $( 1, 2)( 3,34)( 4,33)( 5,31, 6,32)( 7,29, 8,30)( 9,27)(10,28)(11,26,12,25)(13,24,14,23)(15,21,16,22)(17,19,18,20)$ |
| 4H | $4^{4},2^{9}$ | $272$ | $4$ | $21$ | $( 1, 2)( 3,34)( 4,33)( 5,31, 6,32)( 7,29)( 8,30)( 9,27,10,28)(11,25)(12,26)(13,23,14,24)(15,21,16,22)(17,20)(18,19)$ |
| 4I | $4^{3},2^{10},1^{2}$ | $272$ | $4$ | $19$ | $( 3,34)( 4,33)( 5,31, 6,32)( 7,29)( 8,30)( 9,27)(10,28)(11,25,12,26)(13,24,14,23)(15,21)(16,22)(17,20)(18,19)$ |
| 4J | $4^{4},2^{9}$ | $272$ | $4$ | $21$ | $( 1, 2)( 3,34)( 4,33)( 5,32)( 6,31)( 7,29, 8,30)( 9,28,10,27)(11,25,12,26)(13,23)(14,24)(15,21)(16,22)(17,20,18,19)$ |
| 4K | $4^{3},2^{10},1^{2}$ | $272$ | $4$ | $19$ | $( 3,33, 4,34)( 5,32)( 6,31)( 7,29, 8,30)( 9,28)(10,27)(11,26)(12,25)(13,23,14,24)(15,22)(16,21)(17,19)(18,20)$ |
| 4L | $4^{6},2^{5}$ | $272$ | $4$ | $23$ | $( 1, 2)( 3,33, 4,34)( 5,32)( 6,31)( 7,29, 8,30)( 9,28,10,27)(11,26,12,25)(13,24,14,23)(15,22,16,21)(17,19)(18,20)$ |
| 4M | $4^{5},2^{6},1^{2}$ | $272$ | $4$ | $21$ | $( 3,34)( 4,33)( 5,31, 6,32)( 7,29, 8,30)( 9,27,10,28)(11,26)(12,25)(13,23,14,24)(15,21)(16,22)(17,19,18,20)$ |
| 4N | $4^{5},2^{6},1^{2}$ | $272$ | $4$ | $21$ | $( 3,33, 4,34)( 5,32)( 6,31)( 7,29)( 8,30)( 9,28,10,27)(11,25,12,26)(13,24,14,23)(15,22)(16,21)(17,20,18,19)$ |
| 4O | $4^{3},2^{10},1^{2}$ | $272$ | $4$ | $19$ | $( 3,34)( 4,33)( 5,32)( 6,31)( 7,29)( 8,30)( 9,28,10,27)(11,26,12,25)(13,23)(14,24)(15,21,16,22)(17,19)(18,20)$ |
| 17A1 | $17^{2}$ | $512$ | $17$ | $32$ | $( 1,13,25, 4,15,27, 5,18,29, 7,20,32, 9,21,34,12,24)( 2,14,26, 3,16,28, 6,17,30, 8,19,31,10,22,33,11,23)$ |
| 17A2 | $17^{2}$ | $512$ | $17$ | $32$ | $( 1,25,15, 5,29,20, 9,34,24,13, 4,27,18, 7,32,21,12)( 2,26,16, 6,30,19,10,33,23,14, 3,28,17, 8,31,22,11)$ |
| 17A3 | $17^{2}$ | $512$ | $17$ | $32$ | $( 1, 4, 5, 7, 9,12,13,15,18,20,21,24,25,27,29,32,34)( 2, 3, 6, 8,10,11,14,16,17,19,22,23,26,28,30,31,33)$ |
| 17A4 | $17^{2}$ | $512$ | $17$ | $32$ | $( 1,15,29, 9,24, 4,18,32,12,25, 5,20,34,13,27, 7,21)( 2,16,30,10,23, 3,17,31,11,26, 6,19,33,14,28, 8,22)$ |
| 17A5 | $17^{2}$ | $512$ | $17$ | $32$ | $( 1,27,20,12, 4,29,21,13, 5,32,24,15, 7,34,25,18, 9)( 2,28,19,11, 3,30,22,14, 6,31,23,16, 8,33,26,17,10)$ |
| 17A6 | $17^{2}$ | $512$ | $17$ | $32$ | $( 1, 5, 9,13,18,21,25,29,34, 4, 7,12,15,20,24,27,32)( 2, 6,10,14,17,22,26,30,33, 3, 8,11,16,19,23,28,31)$ |
| 17A7 | $17^{2}$ | $512$ | $17$ | $32$ | $( 1,18,34,15,32,13,29,12,27, 9,25, 7,24, 5,21, 4,20)( 2,17,33,16,31,14,30,11,28,10,26, 8,23, 6,22, 3,19)$ |
| 17A8 | $17^{2}$ | $512$ | $17$ | $32$ | $( 1,29,24,18,12, 5,34,27,21,15, 9, 4,32,25,20,13, 7)( 2,30,23,17,11, 6,33,28,22,16,10, 3,31,26,19,14, 8)$ |
Malle's constant $a(G)$: $1/6$
magma: ConjugacyClasses(G);
Character table
40 x 40 character tablemagma: CharacterTable(G);
Regular extensions
Data not computed