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Magma
magma: G := TransitiveGroup(34, 43);
Group invariants
| Abstract group: | $C_2\times \SOMinus(4,4)$ | magma: IdentifyGroup(G);
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| Order: | $16320=2^{6} \cdot 3 \cdot 5 \cdot 17$ | 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: | no | 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 $t$: | $43$ | 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: | $(1,9,12,34,5,31,7,26,4,29,28,21,19,13,16,2,10,11,33,6,32,8,25,3,30,27,22,20,14,15)(17,18)(23,24)$, $(1,21,12,6)(2,22,11,5)(3,32,13,17)(4,31,14,18)(7,27,10,34)(8,28,9,33)(15,24,20,30)(16,23,19,29)(25,26)$ | magma: Generators(G);
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Low degree resolvents
$\card{(G/N)}$ Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_2^2$ $8160$: $\PSL(2,16):C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 17: $\PSL(2,16):C_2$
Low degree siblings
34T43Siblings 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 | $2^{17}$ | $1$ | $2$ | $17$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)$ |
| 2B | $2^{17}$ | $68$ | $2$ | $17$ | $( 1, 2)( 3,14)( 4,13)( 5,15)( 6,16)( 7, 8)( 9,22)(10,21)(11,25)(12,26)(17,23)(18,24)(19,20)(27,32)(28,31)(29,30)(33,34)$ |
| 2C | $2^{12},1^{10}$ | $68$ | $2$ | $12$ | $( 1, 4)( 2, 3)( 9,27)(10,28)(13,26)(14,25)(15,23)(16,24)(17,30)(18,29)(31,34)(32,33)$ |
| 2D | $2^{16},1^{2}$ | $255$ | $2$ | $16$ | $( 1,25)( 2,26)( 3,27)( 4,28)( 5,30)( 6,29)( 7,16)( 8,15)( 9,18)(10,17)(11,20)(12,19)(13,21)(14,22)(23,31)(24,32)$ |
| 2E | $2^{17}$ | $255$ | $2$ | $17$ | $( 1,31)( 2,32)( 3,25)( 4,26)( 5,29)( 6,30)( 7,27)( 8,28)( 9,12)(10,11)(13,24)(14,23)(15,33)(16,34)(17,21)(18,22)(19,20)$ |
| 3A | $3^{10},1^{4}$ | $272$ | $3$ | $20$ | $( 1,30, 7)( 2,29, 8)( 3,27,11)( 4,28,12)( 5,24,10)( 6,23, 9)(13,31,26)(14,32,25)(15,18,21)(16,17,22)$ |
| 4A | $4^{8},1^{2}$ | $1020$ | $4$ | $24$ | $( 1,22,25,14)( 2,21,26,13)( 3,31,27,23)( 4,32,28,24)( 5,10,30,17)( 6, 9,29,18)( 7,19,16,12)( 8,20,15,11)$ |
| 4B | $4^{8},2$ | $1020$ | $4$ | $25$ | $( 1, 8,19,21)( 2, 7,20,22)( 3,17,15,10)( 4,18,16, 9)( 5,27,32,34)( 6,28,31,33)(11,25,13,30)(12,26,14,29)(23,24)$ |
| 5A1 | $5^{6},1^{4}$ | $272$ | $5$ | $24$ | $( 1,17,33,25,10)( 2,18,34,26, 9)( 3,29,31,13,27)( 4,30,32,14,28)( 5, 7,22,12,19)( 6, 8,21,11,20)$ |
| 5A2 | $5^{6},1^{4}$ | $272$ | $5$ | $24$ | $( 1,33,10,17,25)( 2,34, 9,18,26)( 3,31,27,29,13)( 4,32,28,30,14)( 5,22,19, 7,12)( 6,21,20, 8,11)$ |
| 6A | $6^{5},2^{2}$ | $272$ | $6$ | $27$ | $( 1,23, 7, 2,24, 8)( 3,30,20, 4,29,19)( 5, 6)( 9,14,27,10,13,28)(11,33,15,12,34,16)(17,21,32,18,22,31)(25,26)$ |
| 6B | $6^{5},2^{2}$ | $1360$ | $6$ | $27$ | $( 1, 8,30, 2, 7,29)( 3,25,27,14,11,32)( 4,26,28,13,12,31)( 5,21,24,15,10,18)( 6,22,23,16, 9,17)(19,20)(33,34)$ |
| 6C | $6^{4},3^{2},1^{4}$ | $1360$ | $6$ | $24$ | $( 1,10,32, 7,28,22)( 2, 9,31, 8,27,21)( 3,34,13)( 4,33,14)( 5,12,30,25,16,19)( 6,11,29,26,15,20)$ |
| 10A1 | $10^{3},2^{2}$ | $272$ | $10$ | $29$ | $( 1,20,10,11,32, 2,19, 9,12,31)( 3,14,34,17,23, 4,13,33,18,24)( 5, 6)( 7,29,28,15,22, 8,30,27,16,21)(25,26)$ |
| 10A3 | $10^{3},2^{2}$ | $272$ | $10$ | $29$ | $( 1,11,19,31,10, 2,12,20,32, 9)( 3,17,13,24,34, 4,18,14,23,33)( 5, 6)( 7,15,30,21,28, 8,16,29,22,27)(25,26)$ |
| 10B1 | $10^{2},5^{2},2^{2}$ | $816$ | $10$ | $28$ | $( 1,14,17,28,33, 4,25,30,10,32)( 2,13,18,27,34, 3,26,29, 9,31)( 5,12, 7,19,22)( 6,11, 8,20,21)(15,23)(16,24)$ |
| 10B3 | $10^{2},5^{2},2^{2}$ | $816$ | $10$ | $28$ | $( 1,28,25,32,17, 4,10,14,33,30)( 2,27,26,31,18, 3, 9,13,34,29)( 5,19,12,22, 7)( 6,20,11,21, 8)(15,23)(16,24)$ |
| 10C1 | $10^{3},2^{2}$ | $816$ | $10$ | $29$ | $( 1,26,17,29,32, 6,24,20,28, 3)( 2,25,18,30,31, 5,23,19,27, 4)( 7,11,10,13,33, 8,12, 9,14,34)(15,22)(16,21)$ |
| 10C3 | $10^{3},2^{2}$ | $816$ | $10$ | $29$ | $( 1,29,24, 3,17, 6,28,26,32,20)( 2,30,23, 4,18, 5,27,25,31,19)( 7,13,12,34,10, 8,14,11,33, 9)(15,22)(16,21)$ |
| 15A1 | $15^{2},1^{4}$ | $544$ | $15$ | $28$ | $( 1,17,16,10, 4, 7,32,33,28,19,24,22,12,14,30)( 2,18,15, 9, 3, 8,31,34,27,20,23,21,11,13,29)$ |
| 15A2 | $15^{2},1^{4}$ | $544$ | $15$ | $28$ | $( 1,16, 4,32,28,24,12,30,17,10, 7,33,19,22,14)( 2,15, 3,31,27,23,11,29,18, 9, 8,34,20,21,13)$ |
| 17A1 | $17^{2}$ | $480$ | $17$ | $32$ | $( 1,22,30,16,14,10, 5,33, 4, 7,12,17,28,19,32,25,24)( 2,21,29,15,13, 9, 6,34, 3, 8,11,18,27,20,31,26,23)$ |
| 17A2 | $17^{2}$ | $480$ | $17$ | $32$ | $( 1,30,14, 5, 4,12,28,32,24,22,16,10,33, 7,17,19,25)( 2,29,13, 6, 3,11,27,31,23,21,15, 9,34, 8,18,20,26)$ |
| 17A3 | $17^{2}$ | $480$ | $17$ | $32$ | $( 1,16, 5, 7,28,25,22,14,33,12,19,24,30,10, 4,17,32)( 2,15, 6, 8,27,26,21,13,34,11,20,23,29, 9, 3,18,31)$ |
| 17A6 | $17^{2}$ | $480$ | $17$ | $32$ | $( 1, 5,28,22,33,19,30, 4,32,16, 7,25,14,12,24,10,17)( 2, 6,27,21,34,20,29, 3,31,15, 8,26,13,11,23, 9,18)$ |
| 30A1 | $30,2^{2}$ | $544$ | $30$ | $31$ | $( 1,27,17,20,16,23,10,21, 4,11, 7,13,32,29,33, 2,28,18,19,15,24, 9,22, 3,12, 8,14,31,30,34)( 5, 6)(25,26)$ |
| 30A7 | $30,2^{2}$ | $544$ | $30$ | $31$ | $( 1,21,33, 9,30,23,32,15,14,20, 7,18,12,27, 4, 2,22,34,10,29,24,31,16,13,19, 8,17,11,28, 3)( 5, 6)(25,26)$ |
| 34A1 | $34$ | $480$ | $34$ | $33$ | $( 1, 8,22,11,30,18,16,27,14,20,10,31, 5,26,33,23, 4, 2, 7,21,12,29,17,15,28,13,19, 9,32, 6,25,34,24, 3)$ |
| 34A3 | $34$ | $480$ | $34$ | $33$ | $( 1,11,16,20, 5,23, 7,29,28, 9,25, 3,22,18,14,31,33, 2,12,15,19, 6,24, 8,30,27,10,26, 4,21,17,13,32,34)$ |
| 34A7 | $34$ | $480$ | $34$ | $33$ | $( 1,27,33,29,32, 8,14,23,17, 6,22,20, 4,15,25,11,10, 2,28,34,30,31, 7,13,24,18, 5,21,19, 3,16,26,12, 9)$ |
| 34A9 | $34$ | $480$ | $34$ | $33$ | $( 1,20, 7, 9,22,31,12, 6,30,26,17,34,16,23,28, 3,14, 2,19, 8,10,21,32,11, 5,29,25,18,33,15,24,27, 4,13)$ |
Malle's constant $a(G)$: $1/12$
magma: ConjugacyClasses(G);
Character table
32 x 32 character tablemagma: CharacterTable(G);
Regular extensions
Data not computed