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Magma
magma: G := TransitiveGroup(34, 38);
Group invariants
| Abstract group: | $D_{17}^2.D_4$ | 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 $t$: | $38$ | 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: | $(1,5,7,8,17,13,11,10)(2,14,3,6,16,4,15,12)(18,28,31,20,32,22,19,30)(21,34,26,27,29,33,24,23)$, $(1,30,5,20)(2,19,4,31)(3,25)(6,26,17,24)(7,32,16,18)(8,21,15,29)(9,27,14,23)(10,33,13,34)(11,22,12,28)$, $(1,21,14,27,10,33,6,22,2,28,15,34,11,23,7,29,3,18,16,24,12,30,8,19,4,25,17,31,13,20,9,26,5,32)$ | magma: Generators(G);
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Low degree resolvents
$\card{(G/N)}$ Galois groups for stem field(s) $2$: $C_2$ x 7 $4$: $C_2^2$ x 7 $8$: $D_{4}$ x 2, $C_2^3$ $16$: $D_4\times C_2$ $32$: 16T44 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 17: None
Low degree siblings
There are no siblings 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 | $2^{8},1^{18}$ | $34$ | $2$ | $8$ | $( 1,16)( 2,15)( 3,14)( 4,13)( 5,12)( 6,11)( 7,10)( 8, 9)$ |
| 2B | $2^{17}$ | $68$ | $2$ | $17$ | $( 1,31)( 2,28)( 3,25)( 4,22)( 5,19)( 6,33)( 7,30)( 8,27)( 9,24)(10,21)(11,18)(12,32)(13,29)(14,26)(15,23)(16,20)(17,34)$ |
| 2C | $2^{17}$ | $68$ | $2$ | $17$ | $( 1,29)( 2,22)( 3,32)( 4,25)( 5,18)( 6,28)( 7,21)( 8,31)( 9,24)(10,34)(11,27)(12,20)(13,30)(14,23)(15,33)(16,26)(17,19)$ |
| 2D | $2^{16},1^{2}$ | $289$ | $2$ | $16$ | $( 1,10)( 2, 9)( 3, 8)( 4, 7)( 5, 6)(11,17)(12,16)(13,15)(19,34)(20,33)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)$ |
| 4A1 | $4^{8},1^{2}$ | $289$ | $4$ | $24$ | $( 1,10,12, 3)( 2,14,11,16)( 4, 5, 9, 8)( 6,13, 7,17)(18,19,23,22)(20,27,21,31)(24,26,34,32)(25,30,33,28)$ |
| 4A-1 | $4^{8},1^{2}$ | $289$ | $4$ | $24$ | $( 1, 3,12,10)( 2,16,11,14)( 4, 8, 9, 5)( 6,17, 7,13)(18,22,23,19)(20,31,21,27)(24,32,34,26)(25,28,33,30)$ |
| 4B | $4^{8},1^{2}$ | $578$ | $4$ | $24$ | $( 1,15,10,13)( 2,11, 9,17)( 3, 7, 8, 4)( 5,16, 6,12)(19,22,34,31)(20,26,33,27)(21,30,32,23)(24,25,29,28)$ |
| 4C | $4^{8},2$ | $1156$ | $4$ | $25$ | $( 1,19,17,29)( 2,26,16,22)( 3,33,15,32)( 4,23,14,25)( 5,30,13,18)( 6,20,12,28)( 7,27,11,21)( 8,34,10,31)( 9,24)$ |
| 4D | $4^{8},2$ | $1156$ | $4$ | $25$ | $( 1,23, 6,25)( 2,20, 5,28)( 3,34, 4,31)( 7,22,17,26)( 8,19,16,29)( 9,33,15,32)(10,30,14,18)(11,27,13,21)(12,24)$ |
| 8A1 | $8^{4},1^{2}$ | $578$ | $8$ | $28$ | $( 1,12,15, 5,10,16,13, 6)( 2, 3,11, 7, 9, 8,17, 4)(19,33,22,27,34,20,31,26)(21,29,30,28,32,24,23,25)$ |
| 8A3 | $8^{4},1^{2}$ | $578$ | $8$ | $28$ | $( 1, 5,13,12,10, 6,15,16)( 2, 7,17, 3, 9, 4,11, 8)(19,27,31,33,34,26,22,20)(21,28,23,29,32,25,30,24)$ |
| 8B1 | $8^{4},1^{2}$ | $578$ | $8$ | $28$ | $( 1,13, 7,10,17, 5,11, 8)( 2, 4, 3,12,16,14,15, 6)(18,28,31,20,32,22,19,30)(21,34,26,27,29,33,24,23)$ |
| 8B-1 | $8^{4},1^{2}$ | $578$ | $8$ | $28$ | $( 1, 8,11, 5,17,10, 7,13)( 2, 6,15,14,16,12, 3, 4)(18,30,19,22,32,20,31,28)(21,23,24,33,29,27,26,34)$ |
| 17A1 | $17,1^{17}$ | $16$ | $17$ | $16$ | $(18,32,29,26,23,20,34,31,28,25,22,19,33,30,27,24,21)$ |
| 17A3 | $17,1^{17}$ | $16$ | $17$ | $16$ | $(18,26,34,25,33,24,32,23,31,22,30,21,29,20,28,19,27)$ |
| 17B1 | $17^{2}$ | $16$ | $17$ | $32$ | $( 1, 4, 7,10,13,16, 2, 5, 8,11,14,17, 3, 6, 9,12,15)(18,26,34,25,33,24,32,23,31,22,30,21,29,20,28,19,27)$ |
| 17B2 | $17^{2}$ | $16$ | $17$ | $32$ | $( 1, 7,13, 2, 8,14, 3, 9,15, 4,10,16, 5,11,17, 6,12)(18,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19)$ |
| 17B3 | $17^{2}$ | $16$ | $17$ | $32$ | $( 1,10, 2,11, 3,12, 4,13, 5,14, 6,15, 7,16, 8,17, 9)(18,25,32,22,29,19,26,33,23,30,20,27,34,24,31,21,28)$ |
| 17B6 | $17^{2}$ | $16$ | $17$ | $32$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17)(18,32,29,26,23,20,34,31,28,25,22,19,33,30,27,24,21)$ |
| 17C1 | $17^{2}$ | $16$ | $17$ | $32$ | $( 1,16,14,12,10, 8, 6, 4, 2,17,15,13,11, 9, 7, 5, 3)(18,32,29,26,23,20,34,31,28,25,22,19,33,30,27,24,21)$ |
| 17C2 | $17^{2}$ | $16$ | $17$ | $32$ | $( 1,14,10, 6, 2,15,11, 7, 3,16,12, 8, 4,17,13, 9, 5)(18,29,23,34,28,22,33,27,21,32,26,20,31,25,19,30,24)$ |
| 17C3 | $17^{2}$ | $16$ | $17$ | $32$ | $( 1,12, 6,17,11, 5,16,10, 4,15, 9, 3,14, 8, 2,13, 7)(18,26,34,25,33,24,32,23,31,22,30,21,29,20,28,19,27)$ |
| 17C6 | $17^{2}$ | $16$ | $17$ | $32$ | $( 1, 6,11,16, 4, 9,14, 2, 7,12,17, 5,10,15, 3, 8,13)(18,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19)$ |
| 17D1 | $17^{2}$ | $32$ | $17$ | $32$ | $( 1,11, 4,14, 7,17,10, 3,13, 6,16, 9, 2,12, 5,15, 8)(18,24,30,19,25,31,20,26,32,21,27,33,22,28,34,23,29)$ |
| 17D2 | $17^{2}$ | $32$ | $17$ | $32$ | $( 1, 4, 7,10,13,16, 2, 5, 8,11,14,17, 3, 6, 9,12,15)(18,30,25,20,32,27,22,34,29,24,19,31,26,21,33,28,23)$ |
| 17D3 | $17^{2}$ | $32$ | $17$ | $32$ | $( 1, 4, 7,10,13,16, 2, 5, 8,11,14,17, 3, 6, 9,12,15)(18,25,32,22,29,19,26,33,23,30,20,27,34,24,31,21,28)$ |
| 17D6 | $17^{2}$ | $32$ | $17$ | $32$ | $( 1,11, 4,14, 7,17,10, 3,13, 6,16, 9, 2,12, 5,15, 8)(18,30,25,20,32,27,22,34,29,24,19,31,26,21,33,28,23)$ |
| 34A1 | $17,2^{8},1$ | $272$ | $34$ | $24$ | $( 1,16)( 2,15)( 3,14)( 4,13)( 5,12)( 6,11)( 7,10)( 8, 9)(18,25,32,22,29,19,26,33,23,30,20,27,34,24,31,21,28)$ |
| 34A3 | $17,2^{8},1$ | $272$ | $34$ | $24$ | $( 1,16)( 2,15)( 3,14)( 4,13)( 5,12)( 6,11)( 7,10)( 8, 9)(18,22,26,30,34,21,25,29,33,20,24,28,32,19,23,27,31)$ |
| 34B1 | $34$ | $272$ | $34$ | $33$ | $( 1,18, 4,26, 7,34,10,25,13,33,16,24, 2,32, 5,23, 8,31,11,22,14,30,17,21, 3,29, 6,20, 9,28,12,19,15,27)$ |
| 34B3 | $34$ | $272$ | $34$ | $33$ | $( 1,26,10,33, 2,23,11,30, 3,20,12,27, 4,34,13,24, 5,31,14,21, 6,28,15,18, 7,25,16,32, 8,22,17,29, 9,19)$ |
| 34B7 | $34$ | $272$ | $34$ | $33$ | $( 1,25, 5,30, 9,18,13,23,17,28, 4,33, 8,21,12,26,16,31, 3,19, 7,24,11,29,15,34, 2,22, 6,27,10,32,14,20)$ |
| 34B9 | $34$ | $272$ | $34$ | $33$ | $( 1,33,11,20, 4,24,14,28, 7,32,17,19,10,23, 3,27,13,31, 6,18,16,22, 9,26, 2,30,12,34, 5,21,15,25, 8,29)$ |
| 34C1 | $34$ | $272$ | $34$ | $33$ | $( 1,19,16,33,14,30,12,27,10,24, 8,21, 6,18, 4,32, 2,29,17,26,15,23,13,20,11,34, 9,31, 7,28, 5,25, 3,22)$ |
| 34C3 | $34$ | $272$ | $34$ | $33$ | $( 1,33,12,24, 6,32,17,23,11,31, 5,22,16,30,10,21, 4,29,15,20, 9,28, 3,19,14,27, 8,18, 2,26,13,34, 7,25)$ |
| 34C7 | $34$ | $272$ | $34$ | $33$ | $( 1,27, 4,23, 7,19,10,32,13,28,16,24, 2,20, 5,33, 8,29,11,25,14,21,17,34, 3,30, 6,26, 9,22,12,18,15,31)$ |
| 34C9 | $34$ | $272$ | $34$ | $33$ | $( 1,24,17,31,16,21,15,28,14,18,13,25,12,32,11,22,10,29, 9,19, 8,26, 7,33, 6,23, 5,30, 4,20, 3,27, 2,34)$ |
Malle's constant $a(G)$: $1/8$
magma: ConjugacyClasses(G);
Character table
38 x 38 character tablemagma: CharacterTable(G);
Regular extensions
Data not computed