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Magma
magma: G := TransitiveGroup(34, 36);
Group action invariants
| Degree $n$: | $34$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $36$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_{17}^2:\OD_{32}$ | ||
| Parity: | $-1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $1$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,34,8,30,13,32,2,31,16,23,9,27,4,25,15,26)(3,28,7,33,5,22,6,19,14,29,10,24,12,18,11,21)(17,20), (1,12,9,16,11,17,3,13)(2,4,5,14,10,8,7,15)(18,19,27,23,25,24,33,20)(21,26,32,29,22,34,28,31) | magma: Generators(G);
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Low degree resolvents
$\card{(G/N)}$ Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_4$ x 2, $C_2^2$ $8$: $C_8$ x 2, $C_4\times C_2$ $16$: $C_8\times C_2$ $32$: $C_{16} : C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 17: None
Low degree siblings
34T34 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 | $2^{8},1^{18}$ | $34$ | $2$ | $8$ | $( 1,10)( 2, 9)( 3, 8)( 4, 7)( 5, 6)(11,17)(12,16)(13,15)$ |
| 2B | $2^{16},1^{2}$ | $289$ | $2$ | $16$ | $( 2,17)( 3,16)( 4,15)( 5,14)( 6,13)( 7,12)( 8,11)( 9,10)(18,27)(19,26)(20,25)(21,24)(22,23)(28,34)(29,33)(30,32)$ |
| 4A1 | $4^{8},1^{2}$ | $289$ | $4$ | $24$ | $( 2, 5,17,14)( 3, 9,16,10)( 4,13,15, 6)( 7, 8,12,11)(18,30,27,32)(19,34,26,28)(20,21,25,24)(22,29,23,33)$ |
| 4A-1 | $4^{8},1^{2}$ | $289$ | $4$ | $24$ | $( 2,14,17, 5)( 3,10,16, 9)( 4, 6,15,13)( 7,11,12, 8)(18,32,27,30)(19,28,26,34)(20,24,25,21)(22,33,23,29)$ |
| 4B | $4^{8},1^{2}$ | $578$ | $4$ | $24$ | $( 2,14,17, 5)( 3,10,16, 9)( 4, 6,15,13)( 7,11,12, 8)(18,32,20,23)(21,27,34,28)(22,31,33,24)(25,26,30,29)$ |
| 8A1 | $8^{4},1^{2}$ | $289$ | $8$ | $28$ | $( 2,10,14,16,17, 9, 5, 3)( 4,11, 6,12,15, 8,13, 7)(18,33,32,23,27,29,30,22)(19,25,28,21,26,20,34,24)$ |
| 8A-1 | $8^{4},1^{2}$ | $289$ | $8$ | $28$ | $( 2,16, 5,10,17, 3,14, 9)( 4,12,13,11,15, 7, 6, 8)(18,23,30,33,27,22,32,29)(19,21,34,25,26,24,28,20)$ |
| 8A3 | $8^{4},1^{2}$ | $289$ | $8$ | $28$ | $( 2, 9,14, 3,17,10, 5,16)( 4, 8, 6, 7,15,11,13,12)(18,29,32,22,27,33,30,23)(19,20,28,24,26,25,34,21)$ |
| 8A-3 | $8^{4},1^{2}$ | $289$ | $8$ | $28$ | $( 2, 3, 5, 9,17,16,14,10)( 4, 7,13, 8,15,12, 6,11)(18,22,30,29,27,23,32,33)(19,24,34,20,26,21,28,25)$ |
| 8B1 | $8^{4},1^{2}$ | $578$ | $8$ | $28$ | $( 2, 9,14, 3,17,10, 5,16)( 4, 8, 6, 7,15,11,13,12)(19,27,31,33,34,26,22,20)(21,28,23,29,32,25,30,24)$ |
| 8B-1 | $8^{4},1^{2}$ | $578$ | $8$ | $28$ | $( 2,16, 5,10,17, 3,14, 9)( 4,12,13,11,15, 7, 6, 8)(18,24,19,26,23,34,22,32)(20,28,27,25,21,30,31,33)$ |
| 16A1 | $16^{2},2$ | $578$ | $16$ | $31$ | $( 1,33,11,32, 6,24,17,28, 3,26,10,27,15,18, 4,31)( 2,21)( 5,19, 9,22, 7,29, 8,34,16,23,12,20,14,30,13,25)$ |
| 16A-1 | $16^{2},2$ | $578$ | $16$ | $31$ | $( 1,31)( 2,34, 9,21,14,19, 3,20,17,28,10,24, 5,26,16,25)( 4,23, 8,18, 6,29, 7,32,15,22,11,27,13,33,12,30)$ |
| 16A3 | $16^{2},2$ | $578$ | $16$ | $31$ | $( 1,31)( 2,26,10,20,14,34,16,24,17,19, 9,25, 5,28, 3,21)( 4,33,11,32, 6,23,12,27,15,29, 8,30,13,22, 7,18)$ |
| 16A-3 | $16^{2},2$ | $578$ | $16$ | $31$ | $( 1,31)( 2,24, 3,34, 5,20, 9,26,17,21,16,28,14,25,10,19)( 4,27, 7,23,13,32, 8,33,15,18,12,22, 6,30,11,29)$ |
| 16B1 | $16^{2},2$ | $578$ | $16$ | $31$ | $( 1,33,13,20, 3,28,17,27,11,25,16,21, 9,30,12,31)( 2,22,15,32, 7,18, 8,24,10,19,14,26, 5,23, 4,34)( 6,29)$ |
| 16B-1 | $16^{2},2$ | $578$ | $16$ | $31$ | $( 1,33, 4,29,15,20,10,21, 3,19,17,23, 6,32,11,31)( 2,26)( 5,22,13,34,14,27,12,24,16,30, 8,18, 7,25, 9,28)$ |
| 16B3 | $16^{2},2$ | $578$ | $16$ | $31$ | $( 1,31)( 2,20,16,19, 5,21,10,34,17,25, 3,26,14,24, 9,28)( 4,32,12,29,13,18,11,23,15,30, 7,33, 6,27, 8,22)$ |
| 16B-3 | $16^{2},2$ | $578$ | $16$ | $31$ | $( 1,33, 7,34,10,26, 3,22, 8,20, 2,19,16,27, 6,31)( 4,25,17,30,15,24,14,21, 5,28, 9,23,11,29,12,32)(13,18)$ |
| 17A1 | $17,1^{17}$ | $16$ | $17$ | $16$ | $(18,31,27,23,19,32,28,24,20,33,29,25,21,34,30,26,22)$ |
| 17A3 | $17,1^{17}$ | $16$ | $17$ | $16$ | $( 1,16,14,12,10, 8, 6, 4, 2,17,15,13,11, 9, 7, 5, 3)$ |
| 17B | $17^{2}$ | $32$ | $17$ | $32$ | $( 1,15,12, 9, 6, 3,17,14,11, 8, 5, 2,16,13,10, 7, 4)(18,32,29,26,23,20,34,31,28,25,22,19,33,30,27,24,21)$ |
| 17C | $17^{2}$ | $32$ | $17$ | $32$ | $( 1, 4, 7,10,13,16, 2, 5, 8,11,14,17, 3, 6, 9,12,15)(18,30,25,20,32,27,22,34,29,24,19,31,26,21,33,28,23)$ |
| 17D1 | $17^{2}$ | $32$ | $17$ | $32$ | $( 1, 5, 9,13,17, 4, 8,12,16, 3, 7,11,15, 2, 6,10,14)(18,24,30,19,25,31,20,26,32,21,27,33,22,28,34,23,29)$ |
| 17D3 | $17^{2}$ | $32$ | $17$ | $32$ | $( 1,14,10, 6, 2,15,11, 7, 3,16,12, 8, 4,17,13, 9, 5)(18,25,32,22,29,19,26,33,23,30,20,27,34,24,31,21,28)$ |
| 17E1 | $17^{2}$ | $32$ | $17$ | $32$ | $( 1,17,16,15,14,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(18,20,22,24,26,28,30,32,34,19,21,23,25,27,29,31,33)$ |
| 17E3 | $17^{2}$ | $32$ | $17$ | $32$ | $( 1, 6,11,16, 4, 9,14, 2, 7,12,17, 5,10,15, 3, 8,13)(18,22,26,30,34,21,25,29,33,20,24,28,32,19,23,27,31)$ |
| 17F1 | $17^{2}$ | $32$ | $17$ | $32$ | $( 1,13, 8, 3,15,10, 5,17,12, 7, 2,14, 9, 4,16,11, 6)(18,27,19,28,20,29,21,30,22,31,23,32,24,33,25,34,26)$ |
| 17F3 | $17^{2}$ | $32$ | $17$ | $32$ | $( 1, 3, 5, 7, 9,11,13,15,17, 2, 4, 6, 8,10,12,14,16)(18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34)$ |
| 34A1 | $17,2^{8},1$ | $272$ | $34$ | $24$ | $( 2,17)( 3,16)( 4,15)( 5,14)( 6,13)( 7,12)( 8,11)( 9,10)(18,20,22,24,26,28,30,32,34,19,21,23,25,27,29,31,33)$ |
| 34A3 | $17,2^{8},1$ | $272$ | $34$ | $24$ | $( 1,15)( 2,14)( 3,13)( 4,12)( 5,11)( 6,10)( 7, 9)(16,17)(18,32,29,26,23,20,34,31,28,25,22,19,33,30,27,24,21)$ |
Malle's constant $a(G)$: $1/8$
magma: ConjugacyClasses(G);
Group invariants
| 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 | ||
| Label: | 9248.u | magma: IdentifyGroup(G);
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| Character table: | 32 x 32 character table |
magma: CharacterTable(G);