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Magma
magma: G := TransitiveGroup(34, 23);
Group invariants
| Abstract group: | $C_{17}^2:\SD_{16}$ | magma: IdentifyGroup(G);
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| Order: | $4624=2^{4} \cdot 17^{2}$ | 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 $t$: | $23$ | 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: | $(1,22,11,30,4,21,14,29,7,20,17,28,10,19,3,27,13,18,6,26,16,34,9,25,2,33,12,24,5,32,15,23,8,31)$, $(1,5,13,12,10,6,15,16)(2,7,17,3,9,4,11,8)(18,24,21,31,26,20,23,30)(19,32,34,33,25,29,27,28)$ | magma: Generators(G);
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Low degree resolvents
$\card{(G/N)}$ Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_2^2$ $8$: $D_{4}$ $16$: $QD_{16}$ 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 | $2^{17}$ | $68$ | $2$ | $17$ | $( 1,27)( 2,21)( 3,32)( 4,26)( 5,20)( 6,31)( 7,25)( 8,19)( 9,30)(10,24)(11,18)(12,29)(13,23)(14,34)(15,28)(16,22)(17,33)$ |
| 2B | $2^{16},1^{2}$ | $289$ | $2$ | $16$ | $( 1,12)( 2,11)( 3,10)( 4, 9)( 5, 8)( 6, 7)(13,17)(14,16)(19,34)(20,33)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)$ |
| 4A | $4^{8},1^{2}$ | $578$ | $4$ | $24$ | $( 1,10,12, 3)( 2,14,11,16)( 4, 5, 9, 8)( 6,13, 7,17)(19,31,34,22)(20,27,33,26)(21,23,32,30)(24,28,29,25)$ |
| 4B | $4^{8},2$ | $1156$ | $4$ | $25$ | $( 1,21,15,19)( 2,33,14,24)( 3,28,13,29)( 4,23,12,34)( 5,18,11,22)( 6,30,10,27)( 7,25, 9,32)( 8,20)(16,31,17,26)$ |
| 8A1 | $8^{4},1^{2}$ | $578$ | $8$ | $28$ | $( 1, 9,10, 8,12, 4, 3, 5)( 2, 7,14,17,11, 6,16,13)(19,27,31,33,34,26,22,20)(21,28,23,29,32,25,30,24)$ |
| 8A-1 | $8^{4},1^{2}$ | $578$ | $8$ | $28$ | $( 1, 4,10, 5,12, 9, 3, 8)( 2, 6,14,13,11, 7,16,17)(19,26,31,20,34,27,22,33)(21,25,23,24,32,28,30,29)$ |
| 17A1 | $17^{2}$ | $8$ | $17$ | $32$ | $( 1, 9,17, 8,16, 7,15, 6,14, 5,13, 4,12, 3,11, 2,10)(18,32,29,26,23,20,34,31,28,25,22,19,33,30,27,24,21)$ |
| 17A2 | $17^{2}$ | $8$ | $17$ | $32$ | $( 1,16,14,12,10, 8, 6, 4, 2,17,15,13,11, 9, 7, 5, 3)(18,23,28,33,21,26,31,19,24,29,34,22,27,32,20,25,30)$ |
| 17A3 | $17^{2}$ | $8$ | $17$ | $32$ | $( 1,13, 8, 3,15,10, 5,17,12, 7, 2,14, 9, 4,16,11, 6)(18,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19)$ |
| 17A4 | $17^{2}$ | $8$ | $17$ | $32$ | $( 1, 8,15, 5,12, 2, 9,16, 6,13, 3,10,17, 7,14, 4,11)(18,26,34,25,33,24,32,23,31,22,30,21,29,20,28,19,27)$ |
| 17A5 | $17^{2}$ | $8$ | $17$ | $32$ | $( 1,11, 4,14, 7,17,10, 3,13, 6,16, 9, 2,12, 5,15, 8)(18,33,31,29,27,25,23,21,19,34,32,30,28,26,24,22,20)$ |
| 17A6 | $17^{2}$ | $8$ | $17$ | $32$ | $( 1,10, 2,11, 3,12, 4,13, 5,14, 6,15, 7,16, 8,17, 9)(18,30,25,20,32,27,22,34,29,24,19,31,26,21,33,28,23)$ |
| 17A7 | $17^{2}$ | $8$ | $17$ | $32$ | $( 1, 8,15, 5,12, 2, 9,16, 6,13, 3,10,17, 7,14, 4,11)(18,33,31,29,27,25,23,21,19,34,32,30,28,26,24,22,20)$ |
| 17A8 | $17^{2}$ | $8$ | $17$ | $32$ | $( 1,17,16,15,14,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(18,29,23,34,28,22,33,27,21,32,26,20,31,25,19,30,24)$ |
| 17B1 | $17^{2}$ | $16$ | $17$ | $32$ | $( 1, 5, 9,13,17, 4, 8,12,16, 3, 7,11,15, 2, 6,10,14)(18,20,22,24,26,28,30,32,34,19,21,23,25,27,29,31,33)$ |
| 17B3 | $17^{2}$ | $16$ | $17$ | $32$ | $( 1,13, 8, 3,15,10, 5,17,12, 7, 2,14, 9, 4,16,11, 6)(18,24,30,19,25,31,20,26,32,21,27,33,22,28,34,23,29)$ |
| 17C1 | $17,1^{17}$ | $16$ | $17$ | $16$ | $(18,28,21,31,24,34,27,20,30,23,33,26,19,29,22,32,25)$ |
| 17C2 | $17^{2}$ | $16$ | $17$ | $32$ | $( 1,16,14,12,10, 8, 6, 4, 2,17,15,13,11, 9, 7, 5, 3)(18,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19)$ |
| 17C3 | $17^{2}$ | $16$ | $17$ | $32$ | $( 1, 8,15, 5,12, 2, 9,16, 6,13, 3,10,17, 7,14, 4,11)(18,30,25,20,32,27,22,34,29,24,19,31,26,21,33,28,23)$ |
| 17C6 | $17^{2}$ | $16$ | $17$ | $32$ | $( 1, 7,13, 2, 8,14, 3, 9,15, 4,10,16, 5,11,17, 6,12)(18,31,27,23,19,32,28,24,20,33,29,25,21,34,30,26,22)$ |
| 17D1 | $17,1^{17}$ | $16$ | $17$ | $16$ | $( 1,15,12, 9, 6, 3,17,14,11, 8, 5, 2,16,13,10, 7, 4)$ |
| 17D2 | $17^{2}$ | $16$ | $17$ | $32$ | $( 1, 3, 5, 7, 9,11,13,15,17, 2, 4, 6, 8,10,12,14,16)(18,29,23,34,28,22,33,27,21,32,26,20,31,25,19,30,24)$ |
| 17D3 | $17^{2}$ | $16$ | $17$ | $32$ | $( 1,10, 2,11, 3,12, 4,13, 5,14, 6,15, 7,16, 8,17, 9)(18,24,30,19,25,31,20,26,32,21,27,33,22,28,34,23,29)$ |
| 17D4 | $17^{2}$ | $16$ | $17$ | $32$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17)(18,27,19,28,20,29,21,30,22,31,23,32,24,33,25,34,26)$ |
| 17D5 | $17^{2}$ | $16$ | $17$ | $32$ | $( 1,10, 2,11, 3,12, 4,13, 5,14, 6,15, 7,16, 8,17, 9)(18,31,27,23,19,32,28,24,20,33,29,25,21,34,30,26,22)$ |
| 17D6 | $17^{2}$ | $16$ | $17$ | $32$ | $( 1,13, 8, 3,15,10, 5,17,12, 7, 2,14, 9, 4,16,11, 6)(18,27,19,28,20,29,21,30,22,31,23,32,24,33,25,34,26)$ |
| 17D7 | $17^{2}$ | $16$ | $17$ | $32$ | $( 1, 4, 7,10,13,16, 2, 5, 8,11,14,17, 3, 6, 9,12,15)(18,28,21,31,24,34,27,20,30,23,33,26,19,29,22,32,25)$ |
| 17D8 | $17^{2}$ | $16$ | $17$ | $32$ | $( 1, 7,13, 2, 8,14, 3, 9,15, 4,10,16, 5,11,17, 6,12)(18,21,24,27,30,33,19,22,25,28,31,34,20,23,26,29,32)$ |
| 34A1 | $34$ | $136$ | $34$ | $33$ | $( 1,22,16,34,14,29,12,24,10,19, 8,31, 6,26, 4,21, 2,33,17,28,15,23,13,18,11,30, 9,25, 7,20, 5,32, 3,27)$ |
| 34A3 | $34$ | $136$ | $34$ | $33$ | $( 1,26, 4,25, 7,24,10,23,13,22,16,21, 2,20, 5,19, 8,18,11,34,14,33,17,32, 3,31, 6,30, 9,29,12,28,15,27)$ |
| 34A5 | $34$ | $136$ | $34$ | $33$ | $( 1,20, 5,30, 9,23,13,33,17,26, 4,19, 8,29,12,22,16,32, 3,25, 7,18,11,28,15,21, 2,31, 6,24,10,34,14,27)$ |
| 34A7 | $34$ | $136$ | $34$ | $33$ | $( 1,24,10,21, 2,18,11,32, 3,29,12,26, 4,23,13,20, 5,34,14,31, 6,28,15,25, 7,22,16,19, 8,33,17,30, 9,27)$ |
| 34A9 | $34$ | $136$ | $34$ | $33$ | $( 1,21, 2,32, 3,26, 4,20, 5,31, 6,25, 7,19, 8,30, 9,24,10,18,11,29,12,23,13,34,14,28,15,22,16,33,17,27)$ |
| 34A11 | $34$ | $136$ | $34$ | $33$ | $( 1,25, 7,23,13,21, 2,19, 8,34,14,32, 3,30, 9,28,15,26, 4,24,10,22,16,20, 5,18,11,33,17,31, 6,29,12,27)$ |
| 34A13 | $34$ | $136$ | $34$ | $33$ | $( 1,18,11,26, 4,34,14,25, 7,33,17,24,10,32, 3,23,13,31, 6,22,16,30, 9,21, 2,29,12,20, 5,28,15,19, 8,27)$ |
| 34A15 | $34$ | $136$ | $34$ | $33$ | $( 1,23,13,19, 8,32, 3,28,15,24,10,20, 5,33,17,29,12,25, 7,21, 2,34,14,30, 9,26, 4,22,16,18,11,31, 6,27)$ |
Malle's constant $a(G)$: $1/16$
magma: ConjugacyClasses(G);
Character table
37 x 37 character tablemagma: CharacterTable(G);
Regular extensions
Data not computed