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Magma
magma: G := TransitiveGroup(34, 35);
Group invariants
| Abstract group: | $D_{17}^2.(C_2\times C_4)$ | magma: IdentifyGroup(G);
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| Order: | $9248=2^{5} \cdot 17^{2}$ | 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 $t$: | $35$ | 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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| $\card{\Aut(F/K)}$: | $1$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | $(1,14,16,15,7,11,9,10)(2,5,12,17,6,3,13,8)(18,21,28,33,22,19,29,24)(23,27,25,26,34,30,32,31)$, $(1,5,14,13,15,11,2,3)(4,16,9,6,12,17,7,10)(18,22,30,29,27,23,32,33)(19,24,34,20,26,21,28,25)$, $(1,25,17,19,13,29,14,18)(2,31,4,26,12,23,10,28)(3,20,8,33,11,34,6,21)(5,32,16,30,9,22,15,24)(7,27)$ | magma: Generators(G);
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Low degree resolvents
$\card{(G/N)}$ Galois groups for stem field(s) $2$: $C_2$ x 7 $4$: $C_4$ x 4, $C_2^2$ x 7 $8$: $C_4\times C_2$ x 6, $C_2^3$ $16$: $C_4\times C_2^2$ $32$: $(C_8:C_2):C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 17: None
Low degree siblings
34T35 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 | $2^{17}$ | $34$ | $2$ | $17$ | $( 1,23)( 2,18)( 3,30)( 4,25)( 5,20)( 6,32)( 7,27)( 8,22)( 9,34)(10,29)(11,24)(12,19)(13,31)(14,26)(15,21)(16,33)(17,28)$ |
| 2B | $2^{8},1^{18}$ | $34$ | $2$ | $8$ | $( 1,11)( 2,10)( 3, 9)( 4, 8)( 5, 7)(12,17)(13,16)(14,15)$ |
| 2C | $2^{17}$ | $34$ | $2$ | $17$ | $( 1,32)( 2,18)( 3,21)( 4,24)( 5,27)( 6,30)( 7,33)( 8,19)( 9,22)(10,25)(11,28)(12,31)(13,34)(14,20)(15,23)(16,26)(17,29)$ |
| 2D | $2^{16},1^{2}$ | $289$ | $2$ | $16$ | $( 1, 8)( 2, 7)( 3, 6)( 4, 5)( 9,17)(10,16)(11,15)(12,14)(18,32)(19,31)(20,30)(21,29)(22,28)(23,27)(24,26)(33,34)$ |
| 4A1 | $4^{8},1^{2}$ | $289$ | $4$ | $24$ | $( 1,16, 8,10)( 2, 3, 7, 6)( 4,11, 5,15)( 9,14,17,12)(18,31,32,19)(20,22,30,28)(21,26,29,24)(23,34,27,33)$ |
| 4A-1 | $4^{8},1^{2}$ | $289$ | $4$ | $24$ | $( 1,10, 8,16)( 2, 6, 7, 3)( 4,15, 5,11)( 9,12,17,14)(18,19,32,31)(20,28,30,22)(21,24,29,26)(23,33,27,34)$ |
| 4B | $4^{8},2$ | $578$ | $4$ | $25$ | $( 1,31)( 2,26,17,19)( 3,21,16,24)( 4,33,15,29)( 5,28,14,34)( 6,23,13,22)( 7,18,12,27)( 8,30,11,32)( 9,25,10,20)$ |
| 4C | $4^{8},2$ | $578$ | $4$ | $25$ | $( 1,28, 9,18)( 2,31, 8,32)( 3,34, 7,29)( 4,20, 6,26)( 5,23)(10,21,17,25)(11,24,16,22)(12,27,15,19)(13,30,14,33)$ |
| 4D | $4^{8},1^{2}$ | $578$ | $4$ | $24$ | $( 1,17, 4, 5)( 2,13, 3, 9)( 6,14,16, 8)( 7,10,15,12)(18,29,22,28)(19,33,21,24)(23,32,34,25)(26,27,31,30)$ |
| 8A1 | $8^{4},1^{2}$ | $289$ | $8$ | $28$ | $( 1, 3,16, 7, 8, 6,10, 2)( 4,14,11,17, 5,12,15, 9)(18,22,31,30,32,28,19,20)(21,33,26,23,29,34,24,27)$ |
| 8A-1 | $8^{4},1^{2}$ | $289$ | $8$ | $28$ | $( 1, 2,10, 6, 8, 7,16, 3)( 4, 9,15,12, 5,17,11,14)(18,20,19,28,32,30,31,22)(21,27,24,34,29,23,26,33)$ |
| 8A3 | $8^{4},1^{2}$ | $289$ | $8$ | $28$ | $( 1, 7,10, 3, 8, 2,16, 6)( 4,17,15,14, 5, 9,11,12)(18,30,19,22,32,20,31,28)(21,23,24,33,29,27,26,34)$ |
| 8A-3 | $8^{4},1^{2}$ | $289$ | $8$ | $28$ | $( 1, 6,16, 2, 8, 3,10, 7)( 4,12,11, 9, 5,14,15,17)(18,28,31,20,32,22,19,30)(21,34,26,27,29,33,24,23)$ |
| 8B1 | $8^{4},2$ | $578$ | $8$ | $29$ | $( 1,33,12,18, 2,27, 8,25)( 3,21, 4,32,17,22,16,28)( 5,26,13,29,15,34, 7,31)( 6,20, 9,19,14,23,11,24)(10,30)$ |
| 8B-1 | $8^{4},2$ | $578$ | $8$ | $29$ | $( 1,25, 8,27, 2,18,12,33)( 3,28,16,22,17,32, 4,21)( 5,31, 7,34,15,29,13,26)( 6,24,11,23,14,19, 9,20)(10,30)$ |
| 8C1 | $8^{4},2$ | $578$ | $8$ | $29$ | $( 1,21,15,25,10,26,13,22)( 2,31,11,19, 9,33,17,28)( 3,24, 7,30, 8,23, 4,34)( 5,27,16,18, 6,20,12,29)(14,32)$ |
| 8C-1 | $8^{4},2$ | $578$ | $8$ | $29$ | $( 1,22,13,26,10,25,15,21)( 2,28,17,33, 9,19,11,31)( 3,34, 4,23, 8,30, 7,24)( 5,29,12,20, 6,18,16,27)(14,32)$ |
| 8D1 | $8^{4},1^{2}$ | $578$ | $8$ | $28$ | $( 1,14, 6, 7, 9,13, 4, 3)( 2,16,10,15, 8,11,17,12)(18,29,24,34,31,20,25,32)(19,27,28,26,30,22,21,23)$ |
| 8D-1 | $8^{4},1^{2}$ | $578$ | $8$ | $28$ | $( 1, 3, 4,13, 9, 7, 6,14)( 2,12,17,11, 8,15,10,16)(18,32,25,20,31,34,24,29)(19,23,21,22,30,26,28,27)$ |
| 17A1 | $17^{2}$ | $16$ | $17$ | $32$ | $( 1,11, 4,14, 7,17,10, 3,13, 6,16, 9, 2,12, 5,15, 8)(18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34)$ |
| 17A3 | $17^{2}$ | $16$ | $17$ | $32$ | $( 1,14,10, 6, 2,15,11, 7, 3,16,12, 8, 4,17,13, 9, 5)(18,21,24,27,30,33,19,22,25,28,31,34,20,23,26,29,32)$ |
| 17B1 | $17,1^{17}$ | $16$ | $17$ | $16$ | $(18,27,19,28,20,29,21,30,22,31,23,32,24,33,25,34,26)$ |
| 17B3 | $17,1^{17}$ | $16$ | $17$ | $16$ | $(18,28,21,31,24,34,27,20,30,23,33,26,19,29,22,32,25)$ |
| 17C1 | $17^{2}$ | $16$ | $17$ | $32$ | $( 1,12, 6,17,11, 5,16,10, 4,15, 9, 3,14, 8, 2,13, 7)(18,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19)$ |
| 17C3 | $17^{2}$ | $16$ | $17$ | $32$ | $( 1,17,16,15,14,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(18,32,29,26,23,20,34,31,28,25,22,19,33,30,27,24,21)$ |
| 17D1 | $17^{2}$ | $32$ | $17$ | $32$ | $( 1, 6,11,16, 4, 9,14, 2, 7,12,17, 5,10,15, 3, 8,13)(18,32,29,26,23,20,34,31,28,25,22,19,33,30,27,24,21)$ |
| 17D3 | $17^{2}$ | $32$ | $17$ | $32$ | $( 1,16,14,12,10, 8, 6, 4, 2,17,15,13,11, 9, 7, 5, 3)(18,26,34,25,33,24,32,23,31,22,30,21,29,20,28,19,27)$ |
| 17E1 | $17^{2}$ | $32$ | $17$ | $32$ | $( 1,15,12, 9, 6, 3,17,14,11, 8, 5, 2,16,13,10, 7, 4)(18,21,24,27,30,33,19,22,25,28,31,34,20,23,26,29,32)$ |
| 17E3 | $17^{2}$ | $32$ | $17$ | $32$ | $( 1, 7,13, 2, 8,14, 3, 9,15, 4,10,16, 5,11,17, 6,12)(18,32,29,26,23,20,34,31,28,25,22,19,33,30,27,24,21)$ |
| 17F1 | $17^{2}$ | $32$ | $17$ | $32$ | $( 1, 7,13, 2, 8,14, 3, 9,15, 4,10,16, 5,11,17, 6,12)(18,20,22,24,26,28,30,32,34,19,21,23,25,27,29,31,33)$ |
| 17F3 | $17^{2}$ | $32$ | $17$ | $32$ | $( 1, 3, 5, 7, 9,11,13,15,17, 2, 4, 6, 8,10,12,14,16)(18,23,28,33,21,26,31,19,24,29,34,22,27,32,20,25,30)$ |
| 34A1 | $34$ | $272$ | $34$ | $33$ | $( 1,32,11,33, 4,34,14,18, 7,19,17,20,10,21, 3,22,13,23, 6,24,16,25, 9,26, 2,27,12,28, 5,29,15,30, 8,31)$ |
| 34A3 | $34$ | $272$ | $34$ | $33$ | $( 1,33,14,19,10,22, 6,25, 2,28,15,31,11,34, 7,20, 3,23,16,26,12,29, 8,32, 4,18,17,21,13,24, 9,27, 5,30)$ |
| 34B1 | $17,2^{8},1$ | $272$ | $34$ | $24$ | $( 1,11)( 2,10)( 3, 9)( 4, 8)( 5, 7)(12,17)(13,16)(14,15)(18,31,27,23,19,32,28,24,20,33,29,25,21,34,30,26,22)$ |
| 34B3 | $17,2^{8},1$ | $272$ | $34$ | $24$ | $( 1,11)( 2,10)( 3, 9)( 4, 8)( 5, 7)(12,17)(13,16)(14,15)(18,23,28,33,21,26,31,19,24,29,34,22,27,32,20,25,30)$ |
| 34C1 | $34$ | $272$ | $34$ | $33$ | $( 1,23,12,22, 6,21,17,20,11,19, 5,18,16,34,10,33, 4,32,15,31, 9,30, 3,29,14,28, 8,27, 2,26,13,25, 7,24)$ |
| 34C3 | $34$ | $272$ | $34$ | $33$ | $( 1,22,17,19,16,33,15,30,14,27,13,24,12,21,11,18,10,32, 9,29, 8,26, 7,23, 6,20, 5,34, 4,31, 3,28, 2,25)$ |
Malle's constant $a(G)$: $1/8$
magma: ConjugacyClasses(G);
Character table
38 x 38 character tablemagma: CharacterTable(G);
Regular extensions
Data not computed