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Magma
magma: G := TransitiveGroup(34, 46);
Group invariants
Abstract group: | $C_2^8.C_{17}:C_4$ | magma: IdentifyGroup(G);
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Order: | $17408=2^{10} \cdot 17$ | 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 $t$: | $46$ | 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: | $(1,16,28,13)(2,15,27,14)(3,7,26,21,4,8,25,22)(5,33,23,29)(6,34,24,30)(9,18,19,11,10,17,20,12)$, $(1,24)(2,23)(3,22)(4,21)(5,20,6,19)(7,17,8,18)(9,16,10,15)(11,14,12,13)(25,34)(26,33)(27,32,28,31)(29,30)$ | magma: Generators(G);
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Low degree resolvents
$\card{(G/N)}$ Galois groups for stem field(s) $2$: $C_2$ $4$: $C_4$ $68$: $C_{17}:C_{4}$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 17: $C_{17}:C_{4}$
Low degree siblings
34T46 x 2, 34T47 x 3Siblings 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 | $2^{8},1^{18}$ | $17$ | $2$ | $8$ | $( 5, 6)( 7, 8)( 9,10)(13,14)(25,26)(29,30)(31,32)(33,34)$ |
2B | $2^{12},1^{10}$ | $17$ | $2$ | $12$ | $( 5, 6)( 7, 8)(13,14)(15,16)(17,18)(19,20)(21,22)(25,26)(27,28)(29,30)(31,32)(33,34)$ |
2C | $2^{12},1^{10}$ | $17$ | $2$ | $12$ | $( 3, 4)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(21,22)(25,26)(27,28)(31,32)(33,34)$ |
2D | $2^{8},1^{18}$ | $34$ | $2$ | $8$ | $( 1, 2)( 7, 8)( 9,10)(17,18)(19,20)(25,26)(29,30)(31,32)$ |
2E | $2^{10},1^{14}$ | $34$ | $2$ | $10$ | $( 3, 4)( 5, 6)( 9,10)(11,12)(13,14)(15,16)(27,28)(29,30)(31,32)(33,34)$ |
2F | $2^{6},1^{22}$ | $34$ | $2$ | $6$ | $( 5, 6)( 7, 8)(17,18)(21,22)(25,26)(29,30)$ |
2G | $2^{8},1^{18}$ | $34$ | $2$ | $8$ | $( 1, 2)( 5, 6)( 9,10)(13,14)(15,16)(21,22)(27,28)(33,34)$ |
2H | $2^{6},1^{22}$ | $34$ | $2$ | $6$ | $( 9,10)(15,16)(17,18)(19,20)(21,22)(27,28)$ |
2I | $2^{10},1^{14}$ | $34$ | $2$ | $10$ | $( 1, 2)( 7, 8)( 9,10)(13,14)(15,16)(17,18)(25,26)(27,28)(29,30)(33,34)$ |
2J | $2^{16},1^{2}$ | $272$ | $2$ | $16$ | $( 1,28)( 2,27)( 3,26)( 4,25)( 5,23)( 6,24)( 7,22)( 8,21)( 9,19)(10,20)(11,17)(12,18)(13,15)(14,16)(29,33)(30,34)$ |
4A | $4^{6},2^{4},1^{2}$ | $272$ | $4$ | $22$ | $( 1,27, 2,28)( 3,26, 4,25)( 5,23, 6,24)( 7,21, 8,22)( 9,19)(10,20)(11,18)(12,17)(13,16,14,15)(29,33,30,34)$ |
4B | $4^{6},2^{4},1^{2}$ | $272$ | $4$ | $22$ | $( 1,27, 2,28)( 3,26)( 4,25)( 5,23, 6,24)( 7,22)( 8,21)( 9,19,10,20)(11,18,12,17)(13,15,14,16)(29,33,30,34)$ |
4C | $4^{4},2^{8},1^{2}$ | $272$ | $4$ | $20$ | $( 1,28)( 2,27)( 3,26, 4,25)( 5,23)( 6,24)( 7,21, 8,22)( 9,19,10,20)(11,17,12,18)(13,16)(14,15)(29,33)(30,34)$ |
4D | $4^{5},2^{7}$ | $544$ | $4$ | $22$ | $( 1,28, 2,27)( 3,26, 4,25)( 5,24)( 6,23)( 7,22, 8,21)( 9,20)(10,19)(11,17,12,18)(13,15)(14,16)(29,34,30,33)(31,32)$ |
4E | $4^{4},2^{8},1^{2}$ | $544$ | $4$ | $20$ | $( 1,28)( 2,27)( 3,26)( 4,25)( 5,23, 6,24)( 7,21, 8,22)( 9,20,10,19)(11,17)(12,18)(13,16)(14,15)(29,33,30,34)$ |
4F | $4^{3},2^{11}$ | $544$ | $4$ | $20$ | $( 1,27)( 2,28)( 3,26)( 4,25)( 5,24, 6,23)( 7,21)( 8,22)( 9,20)(10,19)(11,18,12,17)(13,16,14,15)(29,34)(30,33)(31,32)$ |
4G | $4^{3},2^{11}$ | $544$ | $4$ | $20$ | $( 1,28, 2,27)( 3,26)( 4,25)( 5,24, 6,23)( 7,22, 8,21)( 9,19)(10,20)(11,17)(12,18)(13,15)(14,16)(29,34)(30,33)(31,32)$ |
4H | $4^{4},2^{8},1^{2}$ | $544$ | $4$ | $20$ | $( 1,27, 2,28)( 3,26, 4,25)( 5,23)( 6,24)( 7,22)( 8,21)( 9,20,10,19)(11,18)(12,17)(13,15,14,16)(29,33)(30,34)$ |
4I | $4^{5},2^{7}$ | $544$ | $4$ | $22$ | $( 1,28, 2,27)( 3,26, 4,25)( 5,24, 6,23)( 7,21)( 8,22)( 9,19,10,20)(11,17,12,18)(13,16)(14,15)(29,34)(30,33)(31,32)$ |
4J1 | $4^{8},1^{2}$ | $1088$ | $4$ | $24$ | $( 1,15,27,14)( 2,16,28,13)( 3, 7,25,22)( 4, 8,26,21)( 5,33,23,29)( 6,34,24,30)( 9,17,20,11)(10,18,19,12)$ |
4J-1 | $4^{8},1^{2}$ | $1088$ | $4$ | $24$ | $( 1,14,28,16)( 2,13,27,15)( 3,21,25, 7)( 4,22,26, 8)( 5,29,23,33)( 6,30,24,34)( 9,12,20,17)(10,11,19,18)$ |
8A1 | $8^{2},4^{4},1^{2}$ | $1088$ | $8$ | $26$ | $( 1,16,28,13)( 2,15,27,14)( 3, 7,26,21, 4, 8,25,22)( 5,33,23,29)( 6,34,24,30)( 9,18,19,11,10,17,20,12)$ |
8A-1 | $8^{3},4^{2},2$ | $1088$ | $8$ | $28$ | $( 1,16,27,13, 2,15,28,14)( 3, 7,25,21)( 4, 8,26,22)( 5,34,23,29, 6,33,24,30)( 9,18,20,11,10,17,19,12)(31,32)$ |
8B1 | $8^{3},4^{2},2$ | $1088$ | $8$ | $28$ | $( 1,14,27,16, 2,13,28,15)( 3,21,26, 7, 4,22,25, 8)( 5,30,23,33, 6,29,24,34)( 9,12,20,18)(10,11,19,17)(31,32)$ |
8B-1 | $8^{3},4^{2},2$ | $1088$ | $8$ | $28$ | $( 1,13,27,16, 2,14,28,15)( 3,21,26, 7)( 4,22,25, 8)( 5,30,23,33, 6,29,24,34)( 9,11,20,17,10,12,19,18)(31,32)$ |
8C1 | $8^{2},4^{4},1^{2}$ | $1088$ | $8$ | $26$ | $( 1,13,28,16)( 2,14,27,15)( 3,21,25, 7, 4,22,26, 8)( 5,29,23,33)( 6,30,24,34)( 9,11,20,18,10,12,19,17)$ |
8C-1 | $8^{3},4^{2},2$ | $1088$ | $8$ | $28$ | $( 1,15,28,14, 2,16,27,13)( 3, 7,26,22, 4, 8,25,21)( 5,34,23,29, 6,33,24,30)( 9,17,19,11)(10,18,20,12)(31,32)$ |
17A1 | $17^{2}$ | $1024$ | $17$ | $32$ | $( 1,16,29, 9,23, 4,18,32,12,26, 5,19,34,13,28, 7,22)( 2,15,30,10,24, 3,17,31,11,25, 6,20,33,14,27, 8,21)$ |
17A2 | $17^{2}$ | $1024$ | $17$ | $32$ | $( 1,29,23,18,12, 5,34,28,22,16, 9, 4,32,26,19,13, 7)( 2,30,24,17,11, 6,33,27,21,15,10, 3,31,25,20,14, 8)$ |
17A3 | $17^{2}$ | $1024$ | $17$ | $32$ | $( 1, 9,18,26,34, 7,16,23,32, 5,13,22,29, 4,12,19,28)( 2,10,17,25,33, 8,15,24,31, 6,14,21,30, 3,11,20,27)$ |
17A6 | $17^{2}$ | $1024$ | $17$ | $32$ | $( 1,18,34,16,32,13,29,12,28, 9,26, 7,23, 5,22, 4,19)( 2,17,33,15,31,14,30,11,27,10,25, 8,24, 6,21, 3,20)$ |
Malle's constant $a(G)$: $1/6$
magma: ConjugacyClasses(G);
Character table
32 x 32 character tablemagma: CharacterTable(G);
Regular extensions
Data not computed