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Magma
magma: G := TransitiveGroup(34, 42);
Group invariants
| Abstract group: | $C_{17}^2:C_{32}$ | 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 $t$: | $42$ | 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: | $(1,32,15,31,16,20,10,18,12,30,17,26,4,33,14,25,5,22,8,23,7,34,13,19,11,24,6,28,2,21,9,29)(3,27)$, $(1,17,6,4,16,12,2,11,8,9,3,5,10,14,7,15)(18,26,29,28,34,32,27,23,30,22,19,20,31,33,21,25)$ | magma: Generators(G);
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Low degree resolvents
$\card{(G/N)}$ Galois groups for stem field(s) $2$: $C_2$ $4$: $C_4$ $8$: $C_8$ $16$: $C_{16}$ $32$: 32T33 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 17: None
Low degree siblings
34T42 x 8Siblings 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 | $2^{16},1^{2}$ | $289$ | $2$ | $16$ | $( 1, 3)( 4,17)( 5,16)( 6,15)( 7,14)( 8,13)( 9,12)(10,11)(18,21)(19,20)(22,34)(23,33)(24,32)(25,31)(26,30)(27,29)$ |
| 4A1 | $4^{8},1^{2}$ | $289$ | $4$ | $24$ | $( 1, 6, 3,15)( 4,11,17,10)( 5, 7,16,14)( 8,12,13, 9)(18,34,21,22)(19,30,20,26)(23,31,33,25)(24,27,32,29)$ |
| 4A-1 | $4^{8},1^{2}$ | $289$ | $4$ | $24$ | $( 1,15, 3, 6)( 4,10,17,11)( 5,14,16, 7)( 8, 9,13,12)(18,22,21,34)(19,26,20,30)(23,25,33,31)(24,29,32,27)$ |
| 8A1 | $8^{4},1^{2}$ | $289$ | $8$ | $28$ | $( 1,11, 6,17, 3,10,15, 4)( 5, 9, 7, 8,16,12,14,13)(18,33,34,25,21,23,22,31)(19,24,30,27,20,32,26,29)$ |
| 8A-1 | $8^{4},1^{2}$ | $289$ | $8$ | $28$ | $( 1, 4,15,10, 3,17, 6,11)( 5,13,14,12,16, 8, 7, 9)(18,31,22,23,21,25,34,33)(19,29,26,32,20,27,30,24)$ |
| 8A3 | $8^{4},1^{2}$ | $289$ | $8$ | $28$ | $( 1,17,15,11, 3, 4, 6,10)( 5, 8,14, 9,16,13, 7,12)(18,25,22,33,21,31,34,23)(19,27,26,24,20,29,30,32)$ |
| 8A-3 | $8^{4},1^{2}$ | $289$ | $8$ | $28$ | $( 1,10, 6, 4, 3,11,15,17)( 5,12, 7,13,16, 9,14, 8)(18,23,34,31,21,33,22,25)(19,32,30,29,20,24,26,27)$ |
| 16A1 | $16^{2},1^{2}$ | $289$ | $16$ | $30$ | $( 1, 7,11, 8, 6,16,17,12, 3,14,10,13,15, 5, 4, 9)(18,27,33,20,34,32,25,26,21,29,23,19,22,24,31,30)$ |
| 16A-1 | $16^{2},1^{2}$ | $289$ | $16$ | $30$ | $( 1, 9, 4, 5,15,13,10,14, 3,12,17,16, 6, 8,11, 7)(18,30,31,24,22,19,23,29,21,26,25,32,34,20,33,27)$ |
| 16A3 | $16^{2},1^{2}$ | $289$ | $16$ | $30$ | $( 1, 8,17,14,15, 9,11,16, 3,13, 4, 7, 6,12,10, 5)(18,20,25,29,22,30,33,32,21,19,31,27,34,26,23,24)$ |
| 16A-3 | $16^{2},1^{2}$ | $289$ | $16$ | $30$ | $( 1, 5,10,12, 6, 7, 4,13, 3,16,11, 9,15,14,17, 8)(18,24,23,26,34,27,31,19,21,32,33,30,22,29,25,20)$ |
| 16A5 | $16^{2},1^{2}$ | $289$ | $16$ | $30$ | $( 1,16,10, 9, 6,14, 4, 8, 3, 5,11,12,15, 7,17,13)(18,32,23,30,34,29,31,20,21,24,33,26,22,27,25,19)$ |
| 16A-5 | $16^{2},1^{2}$ | $289$ | $16$ | $30$ | $( 1,13,17, 7,15,12,11, 5, 3, 8, 4,14, 6, 9,10,16)(18,19,25,27,22,26,33,24,21,20,31,29,34,30,23,32)$ |
| 16A7 | $16^{2},1^{2}$ | $289$ | $16$ | $30$ | $( 1,12, 4,16,15, 8,10, 7, 3, 9,17, 5, 6,13,11,14)(18,26,31,32,22,20,23,27,21,30,25,24,34,19,33,29)$ |
| 16A-7 | $16^{2},1^{2}$ | $289$ | $16$ | $30$ | $( 1,14,11,13, 6, 5,17, 9, 3, 7,10, 8,15,16, 4,12)(18,29,33,19,34,24,25,30,21,27,23,20,22,32,31,26)$ |
| 17A | $17^{2}$ | $32$ | $17$ | $32$ | $( 1,17,16,15,14,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(18,30,25,20,32,27,22,34,29,24,19,31,26,21,33,28,23)$ |
| 17B | $17^{2}$ | $32$ | $17$ | $32$ | $( 1,11, 4,14, 7,17,10, 3,13, 6,16, 9, 2,12, 5,15, 8)(18,29,23,34,28,22,33,27,21,32,26,20,31,25,19,30,24)$ |
| 17C | $17^{2}$ | $32$ | $17$ | $32$ | $( 1,10, 2,11, 3,12, 4,13, 5,14, 6,15, 7,16, 8,17, 9)(18,24,30,19,25,31,20,26,32,21,27,33,22,28,34,23,29)$ |
| 17D | $17^{2}$ | $32$ | $17$ | $32$ | $( 1, 9,17, 8,16, 7,15, 6,14, 5,13, 4,12, 3,11, 2,10)(18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34)$ |
| 17E | $17,1^{17}$ | $32$ | $17$ | $16$ | $( 1, 3, 5, 7, 9,11,13,15,17, 2, 4, 6, 8,10,12,14,16)$ |
| 17F | $17^{2}$ | $32$ | $17$ | $32$ | $( 1, 8,15, 5,12, 2, 9,16, 6,13, 3,10,17, 7,14, 4,11)(18,31,27,23,19,32,28,24,20,33,29,25,21,34,30,26,22)$ |
| 17G | $17^{2}$ | $32$ | $17$ | $32$ | $( 1, 7,13, 2, 8,14, 3, 9,15, 4,10,16, 5,11,17, 6,12)(18,26,34,25,33,24,32,23,31,22,30,21,29,20,28,19,27)$ |
| 17H | $17^{2}$ | $32$ | $17$ | $32$ | $( 1,12, 6,17,11, 5,16,10, 4,15, 9, 3,14, 8, 2,13, 7)(18,24,30,19,25,31,20,26,32,21,27,33,22,28,34,23,29)$ |
| 17I | $17^{2}$ | $32$ | $17$ | $32$ | $( 1, 6,11,16, 4, 9,14, 2, 7,12,17, 5,10,15, 3, 8,13)(18,23,28,33,21,26,31,19,24,29,34,22,27,32,20,25,30)$ |
| 32A1 | $32,2$ | $289$ | $32$ | $32$ | $( 1,20, 7,34,11,32, 8,25, 6,26,16,21,17,29,12,23, 3,19,14,22,10,24,13,31,15,30, 5,18, 4,27, 9,33)( 2,28)$ |
| 32A-1 | $32,2$ | $289$ | $32$ | $32$ | $( 1,33, 9,27, 4,18, 5,30,15,31,13,24,10,22,14,19, 3,23,12,29,17,21,16,26, 6,25, 8,32,11,34, 7,20)( 2,28)$ |
| 32A3 | $32,2$ | $289$ | $32$ | $32$ | $( 1,34, 8,26,17,23,14,24,15,18, 9,20,11,25,16,29, 3,22,13,30, 4,33, 7,32, 6,21,12,19,10,31, 5,27)( 2,28)$ |
| 32A-3 | $32,2$ | $289$ | $32$ | $32$ | $( 1,27, 5,31,10,19,12,21, 6,32, 7,33, 4,30,13,22, 3,29,16,25,11,20, 9,18,15,24,14,23,17,26, 8,34)( 2,28)$ |
| 32A5 | $32,2$ | $289$ | $32$ | $32$ | $( 1,32,16,23,10,30, 9,34, 6,29,14,31, 4,20, 8,21, 3,24, 5,33,11,26,12,22,15,27, 7,25,17,19,13,18)( 2,28)$ |
| 32A-5 | $32,2$ | $289$ | $32$ | $32$ | $( 1,18,13,19,17,25, 7,27,15,22,12,26,11,33, 5,24, 3,21, 8,20, 4,31,14,29, 6,34, 9,30,10,23,16,32)( 2,28)$ |
| 32A7 | $32,2$ | $289$ | $32$ | $32$ | $( 1,25,12,24, 4,34,16,19,15,33, 8,29,10,18, 7,26, 3,31, 9,32,17,22, 5,20, 6,23,13,27,11,21,14,30)( 2,28)$ |
| 32A-7 | $32,2$ | $289$ | $32$ | $32$ | $( 1,30,14,21,11,27,13,23, 6,20, 5,22,17,32, 9,31, 3,26, 7,18,10,29, 8,33,15,19,16,34, 4,24,12,25)( 2,28)$ |
| 32A9 | $32,2$ | $289$ | $32$ | $32$ | $( 1,26,14,18,11,29,13,33, 6,19, 5,34,17,24, 9,25, 3,30, 7,21,10,27, 8,23,15,20,16,22, 4,32,12,31)( 2,28)$ |
| 32A-9 | $32,2$ | $289$ | $32$ | $32$ | $( 1,31,12,32, 4,22,16,20,15,23, 8,27,10,21, 7,30, 3,25, 9,24,17,34, 5,19, 6,33,13,29,11,18,14,26)( 2,28)$ |
| 32A11 | $32,2$ | $289$ | $32$ | $32$ | $( 1,21,13,20,17,31, 7,29,15,34,12,30,11,23, 5,32, 3,18, 8,19, 4,25,14,27, 6,22, 9,26,10,33,16,24)( 2,28)$ |
| 32A-11 | $32,2$ | $289$ | $32$ | $32$ | $( 1,24,16,33,10,26, 9,22, 6,27,14,25, 4,19, 8,18, 3,32, 5,23,11,30,12,34,15,29, 7,31,17,20,13,21)( 2,28)$ |
| 32A13 | $32,2$ | $289$ | $32$ | $32$ | $( 1,29, 5,25,10,20,12,18, 6,24, 7,23, 4,26,13,34, 3,27,16,31,11,19, 9,21,15,32,14,33,17,30, 8,22)( 2,28)$ |
| 32A-13 | $32,2$ | $289$ | $32$ | $32$ | $( 1,22, 8,30,17,33,14,32,15,21, 9,19,11,31,16,27, 3,34,13,26, 4,23, 7,24, 6,18,12,20,10,25, 5,29)( 2,28)$ |
| 32A15 | $32,2$ | $289$ | $32$ | $32$ | $( 1,23, 9,29, 4,21, 5,26,15,25,13,32,10,34,14,20, 3,33,12,27,17,18,16,30, 6,31, 8,24,11,22, 7,19)( 2,28)$ |
| 32A-15 | $32,2$ | $289$ | $32$ | $32$ | $( 1,19, 7,22,11,24, 8,31, 6,30,16,18,17,27,12,33, 3,20,14,34,10,32,13,25,15,26, 5,21, 4,29, 9,23)( 2,28)$ |
Malle's constant $a(G)$: $1/16$
magma: ConjugacyClasses(G);
Character table
41 x 41 character tablemagma: CharacterTable(G);
Regular extensions
Data not computed