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Magma
magma: G := TransitiveGroup(34, 47);
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 $t$: | $47$ | 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: | $(1,26,32,8)(2,25,31,7)(3,17,30,15)(4,18,29,16)(5,9,27,23,6,10,28,24)(11,19,22,14,12,20,21,13)(33,34)$, $(1,33,25,28,2,34,26,27)(3,8,23,20)(4,7,24,19)(5,16,21,12,6,15,22,11)(9,32,17,29,10,31,18,30)$ | 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 3, 34T47 x 2Siblings 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 | $2^{12},1^{10}$ | $17$ | $2$ | $12$ | $( 1, 2)( 7, 8)( 9,10)(15,16)(17,18)(19,20)(21,22)(23,24)(27,28)(29,30)(31,32)(33,34)$ |
2B | $2^{8},1^{18}$ | $17$ | $2$ | $8$ | $( 5, 6)( 9,10)(11,12)(13,14)(19,20)(21,22)(23,24)(27,28)$ |
2C | $2^{12},1^{10}$ | $17$ | $2$ | $12$ | $( 1, 2)( 3, 4)( 7, 8)( 9,10)(13,14)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(31,32)$ |
2D | $2^{8},1^{18}$ | $34$ | $2$ | $8$ | $( 1, 2)( 3, 4)( 9,10)(15,16)(21,22)(23,24)(27,28)(31,32)$ |
2E | $2^{10},1^{14}$ | $34$ | $2$ | $10$ | $( 5, 6)( 7, 8)( 9,10)(13,14)(15,16)(21,22)(23,24)(27,28)(29,30)(31,32)$ |
2F | $2^{8},1^{18}$ | $34$ | $2$ | $8$ | $( 1, 2)( 7, 8)(11,12)(13,14)(15,16)(17,18)(21,22)(27,28)$ |
2G | $2^{6},1^{22}$ | $34$ | $2$ | $6$ | $( 3, 4)( 5, 6)( 9,10)(11,12)(19,20)(29,30)$ |
2H | $2^{10},1^{14}$ | $34$ | $2$ | $10$ | $( 1, 2)( 3, 4)( 7, 8)( 9,10)(13,14)(15,16)(17,18)(19,20)(31,32)(33,34)$ |
2I | $2^{6},1^{22}$ | $34$ | $2$ | $6$ | $( 7, 8)( 9,10)(13,14)(21,22)(25,26)(27,28)$ |
2J | $2^{16},1^{2}$ | $272$ | $2$ | $16$ | $( 1,27)( 2,28)( 3,26)( 4,25)( 5,24)( 6,23)( 7,22)( 8,21)( 9,20)(10,19)(11,17)(12,18)(13,16)(14,15)(29,33)(30,34)$ |
4A | $4^{6},2^{4},1^{2}$ | $272$ | $4$ | $22$ | $( 1,16, 2,15)( 3,13)( 4,14)( 5,12)( 6,11)( 7, 9, 8,10)(17,33,18,34)(19,31,20,32)(21,29,22,30)(23,28,24,27)$ |
4B | $4^{4},2^{8},1^{2}$ | $272$ | $4$ | $20$ | $( 1,32)( 2,31)( 3,29)( 4,30)( 5,28, 6,27)( 7,25)( 8,26)( 9,23,10,24)(11,21,12,22)(13,19,14,20)(15,18)(16,17)$ |
4C | $4^{6},2^{4},1^{2}$ | $272$ | $4$ | $22$ | $( 1,10, 2, 9)( 3, 8, 4, 7)(11,34)(12,33)(13,31,14,32)(15,30)(16,29)(17,27,18,28)(19,26,20,25)(21,23,22,24)$ |
4D | $4^{4},2^{8},1^{2}$ | $544$ | $4$ | $20$ | $( 1,24, 2,23)( 3,21, 4,22)( 5,19)( 6,20)( 7,17)( 8,18)( 9,16,10,15)(11,14)(12,13)(25,33)(26,34)(27,31,28,32)$ |
4E | $4^{5},2^{7}$ | $544$ | $4$ | $22$ | $( 1, 2)( 3,33)( 4,34)( 5,31, 6,32)( 7,30, 8,29)( 9,27,10,28)(11,26)(12,25)(13,24,14,23)(15,22,16,21)(17,20)(18,19)$ |
4F | $4^{4},2^{8},1^{2}$ | $544$ | $4$ | $20$ | $( 1,28, 2,27)( 3,26)( 4,25)( 5,23)( 6,24)( 7,22, 8,21)( 9,20)(10,19)(11,18,12,17)(13,15,14,16)(29,34)(30,33)$ |
4G | $4^{3},2^{11}$ | $544$ | $4$ | $20$ | $( 1,13)( 2,14)( 3,11, 4,12)( 5,10, 6, 9)( 7, 8)(15,34)(16,33)(17,31)(18,32)(19,29,20,30)(21,27)(22,28)(23,25)(24,26)$ |
4H | $4^{5},2^{7}$ | $544$ | $4$ | $22$ | $( 1,16, 2,15)( 3,13, 4,14)( 5,12)( 6,11)( 7, 9, 8,10)(17,33,18,34)(19,32,20,31)(21,30)(22,29)(23,27)(24,28)(25,26)$ |
4I | $4^{3},2^{11}$ | $544$ | $4$ | $20$ | $( 1,34)( 2,33)( 3,31)( 4,32)( 5,29)( 6,30)( 7,28, 8,27)( 9,26,10,25)(11,24)(12,23)(13,21,14,22)(15,19)(16,20)(17,18)$ |
4J1 | $4^{8},2$ | $1088$ | $4$ | $25$ | $( 1,16,27,13)( 2,15,28,14)( 3, 7,26,22)( 4, 8,25,21)( 5,33,24,29)( 6,34,23,30)( 9,17,20,11)(10,18,19,12)(31,32)$ |
4J-1 | $4^{8},2$ | $1088$ | $4$ | $25$ | $( 1,13,27,16)( 2,14,28,15)( 3,22,26, 7)( 4,21,25, 8)( 5,29,24,33)( 6,30,23,34)( 9,11,20,17)(10,12,19,18)(31,32)$ |
8A1 | $8^{3},4^{2},1^{2}$ | $1088$ | $8$ | $27$ | $( 1,19,16,31, 2,20,15,32)( 3,12,13, 5)( 4,11,14, 6)( 7,29, 9,22, 8,30,10,21)(17,23,33,28,18,24,34,27)$ |
8A-1 | $8^{3},4^{2},1^{2}$ | $1088$ | $8$ | $27$ | $( 1,32,15,20, 2,31,16,19)( 3, 5,13,12)( 4, 6,14,11)( 7,21,10,30, 8,22, 9,29)(17,27,34,24,18,28,33,23)$ |
8B1 | $8^{2},4^{4},2$ | $1088$ | $8$ | $27$ | $( 1,25,32, 7)( 2,26,31, 8)( 3,18,29,15)( 4,17,30,16)( 5, 9,28,23, 6,10,27,24)(11,20,21,13,12,19,22,14)(33,34)$ |
8B-1 | $8^{2},4^{4},2$ | $1088$ | $8$ | $27$ | $( 1, 7,32,25)( 2, 8,31,26)( 3,15,29,18)( 4,16,30,17)( 5,24,27,10, 6,23,28, 9)(11,14,22,19,12,13,21,20)(33,34)$ |
8C1 | $8^{3},4^{2},1^{2}$ | $1088$ | $8$ | $27$ | $( 1,24,10,21, 2,23, 9,22)( 3,32, 8,13, 4,31, 7,14)(11,30,34,15)(12,29,33,16)(17,19,27,26,18,20,28,25)$ |
8C-1 | $8^{3},4^{2},1^{2}$ | $1088$ | $8$ | $27$ | $( 1,22, 9,23, 2,21,10,24)( 3,14, 7,31, 4,13, 8,32)(11,15,34,30)(12,16,33,29)(17,25,28,20,18,26,27,19)$ |
17A1 | $17^{2}$ | $1024$ | $17$ | $32$ | $( 1,12,22,31, 7,18,27, 4,13,24,34,10,19,29, 5,15,26)( 2,11,21,32, 8,17,28, 3,14,23,33, 9,20,30, 6,16,25)$ |
17A2 | $17^{2}$ | $1024$ | $17$ | $32$ | $( 1,22, 7,27,13,34,19, 5,26,12,31,18, 4,24,10,29,15)( 2,21, 8,28,14,33,20, 6,25,11,32,17, 3,23, 9,30,16)$ |
17A3 | $17^{2}$ | $1024$ | $17$ | $32$ | $( 1,31,27,24,19,15,12, 7, 4,34,29,26,22,18,13,10, 5)( 2,32,28,23,20,16,11, 8, 3,33,30,25,21,17,14, 9, 6)$ |
17A6 | $17^{2}$ | $1024$ | $17$ | $32$ | $( 1,27,19,12, 4,29,22,13, 5,31,24,15, 7,34,26,18,10)( 2,28,20,11, 3,30,21,14, 6,32,23,16, 8,33,25,17, 9)$ |
Malle's constant $a(G)$: $1/6$
magma: ConjugacyClasses(G);
Character table
32 x 32 character tablemagma: CharacterTable(G);
Regular extensions
Data not computed