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Magma
magma: G := TransitiveGroup(34, 19);
Group invariants
| Abstract group: | $C_{17}^2:D_8$ | 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 $t$: | $19$ | 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: | $(1,32,9,34,17,19,8,21,16,23,7,25,15,27,6,29,14,31,5,33,13,18,4,20,12,22,3,24,11,26,2,28,10,30)$, $(1,17,2,15,6,7,5,9)(3,13,10,16,4,11,14,8)(18,34,26,30,28,29,20,33)(19,25,22,32,27,21,24,31)$ | magma: Generators(G);
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Low degree resolvents
$\card{(G/N)}$ Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_2^2$ $8$: $D_{4}$ $16$: $D_{8}$ 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 | $2^{17}$ | $68$ | $2$ | $17$ | $( 1,25)( 2,27)( 3,29)( 4,31)( 5,33)( 6,18)( 7,20)( 8,22)( 9,24)(10,26)(11,28)(12,30)(13,32)(14,34)(15,19)(16,21)(17,23)$ |
| 2B | $2^{17}$ | $68$ | $2$ | $17$ | $( 1,30)( 2,26)( 3,22)( 4,18)( 5,31)( 6,27)( 7,23)( 8,19)( 9,32)(10,28)(11,24)(12,20)(13,33)(14,29)(15,25)(16,21)(17,34)$ |
| 2C | $2^{16},1^{2}$ | $289$ | $2$ | $16$ | $( 1, 8)( 2, 7)( 3, 6)( 4, 5)( 9,17)(10,16)(11,15)(12,14)(18,22)(19,21)(23,34)(24,33)(25,32)(26,31)(27,30)(28,29)$ |
| 4A | $4^{8},1^{2}$ | $578$ | $4$ | $24$ | $( 1,16, 8,10)( 2, 3, 7, 6)( 4,11, 5,15)( 9,14,17,12)(18,28,22,29)(19,24,21,33)(23,25,34,32)(26,30,31,27)$ |
| 8A1 | $8^{4},1^{2}$ | $578$ | $8$ | $28$ | $( 1, 3,16, 7, 8, 6,10, 2)( 4,14,11,17, 5,12,15, 9)(18,21,28,33,22,19,29,24)(23,27,25,26,34,30,32,31)$ |
| 8A3 | $8^{4},1^{2}$ | $578$ | $8$ | $28$ | $( 1, 7,10, 3, 8, 2,16, 6)( 4,17,15,14, 5, 9,11,12)(18,33,29,21,22,24,28,19)(23,26,32,27,34,31,25,30)$ |
| 17A1 | $17^{2}$ | $8$ | $17$ | $32$ | $( 1,16,14,12,10, 8, 6, 4, 2,17,15,13,11, 9, 7, 5, 3)(18,31,27,23,19,32,28,24,20,33,29,25,21,34,30,26,22)$ |
| 17A2 | $17^{2}$ | $8$ | $17$ | $32$ | $( 1,14,10, 6, 2,15,11, 7, 3,16,12, 8, 4,17,13, 9, 5)(18,27,19,28,20,29,21,30,22,31,23,32,24,33,25,34,26)$ |
| 17A3 | $17^{2}$ | $8$ | $17$ | $32$ | $( 1,12, 6,17,11, 5,16,10, 4,15, 9, 3,14, 8, 2,13, 7)(18,23,28,33,21,26,31,19,24,29,34,22,27,32,20,25,30)$ |
| 17A4 | $17^{2}$ | $8$ | $17$ | $32$ | $( 1,10, 2,11, 3,12, 4,13, 5,14, 6,15, 7,16, 8,17, 9)(18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34)$ |
| 17A5 | $17^{2}$ | $8$ | $17$ | $32$ | $( 1, 8,15, 5,12, 2, 9,16, 6,13, 3,10,17, 7,14, 4,11)(18,32,29,26,23,20,34,31,28,25,22,19,33,30,27,24,21)$ |
| 17A6 | $17^{2}$ | $8$ | $17$ | $32$ | $( 1, 6,11,16, 4, 9,14, 2, 7,12,17, 5,10,15, 3, 8,13)(18,28,21,31,24,34,27,20,30,23,33,26,19,29,22,32,25)$ |
| 17A7 | $17^{2}$ | $8$ | $17$ | $32$ | $( 1, 4, 7,10,13,16, 2, 5, 8,11,14,17, 3, 6, 9,12,15)(18,24,30,19,25,31,20,26,32,21,27,33,22,28,34,23,29)$ |
| 17A8 | $17^{2}$ | $8$ | $17$ | $32$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17)(18,20,22,24,26,28,30,32,34,19,21,23,25,27,29,31,33)$ |
| 17B1 | $17^{2}$ | $8$ | $17$ | $32$ | $( 1,15,12, 9, 6, 3,17,14,11, 8, 5, 2,16,13,10, 7, 4)(18,30,25,20,32,27,22,34,29,24,19,31,26,21,33,28,23)$ |
| 17B2 | $17^{2}$ | $8$ | $17$ | $32$ | $( 1,12, 6,17,11, 5,16,10, 4,15, 9, 3,14, 8, 2,13, 7)(18,25,32,22,29,19,26,33,23,30,20,27,34,24,31,21,28)$ |
| 17B3 | $17^{2}$ | $8$ | $17$ | $32$ | $( 1, 9,17, 8,16, 7,15, 6,14, 5,13, 4,12, 3,11, 2,10)(18,20,22,24,26,28,30,32,34,19,21,23,25,27,29,31,33)$ |
| 17B4 | $17^{2}$ | $8$ | $17$ | $32$ | $( 1, 6,11,16, 4, 9,14, 2, 7,12,17, 5,10,15, 3, 8,13)(18,32,29,26,23,20,34,31,28,25,22,19,33,30,27,24,21)$ |
| 17B5 | $17^{2}$ | $8$ | $17$ | $32$ | $( 1, 3, 5, 7, 9,11,13,15,17, 2, 4, 6, 8,10,12,14,16)(18,27,19,28,20,29,21,30,22,31,23,32,24,33,25,34,26)$ |
| 17B6 | $17^{2}$ | $8$ | $17$ | $32$ | $( 1,17,16,15,14,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(18,22,26,30,34,21,25,29,33,20,24,28,32,19,23,27,31)$ |
| 17B7 | $17^{2}$ | $8$ | $17$ | $32$ | $( 1,14,10, 6, 2,15,11, 7, 3,16,12, 8, 4,17,13, 9, 5)(18,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19)$ |
| 17B8 | $17^{2}$ | $8$ | $17$ | $32$ | $( 1,11, 4,14, 7,17,10, 3,13, 6,16, 9, 2,12, 5,15, 8)(18,29,23,34,28,22,33,27,21,32,26,20,31,25,19,30,24)$ |
| 17C1 | $17,1^{17}$ | $16$ | $17$ | $16$ | $(18,22,26,30,34,21,25,29,33,20,24,28,32,19,23,27,31)$ |
| 17C3 | $17,1^{17}$ | $16$ | $17$ | $16$ | $(18,30,25,20,32,27,22,34,29,24,19,31,26,21,33,28,23)$ |
| 17D1 | $17^{2}$ | $16$ | $17$ | $32$ | $( 1,16,14,12,10, 8, 6, 4, 2,17,15,13,11, 9, 7, 5, 3)(18,24,30,19,25,31,20,26,32,21,27,33,22,28,34,23,29)$ |
| 17D2 | $17^{2}$ | $16$ | $17$ | $32$ | $( 1,14,10, 6, 2,15,11, 7, 3,16,12, 8, 4,17,13, 9, 5)(18,30,25,20,32,27,22,34,29,24,19,31,26,21,33,28,23)$ |
| 17D3 | $17^{2}$ | $16$ | $17$ | $32$ | $( 1, 6,11,16, 4, 9,14, 2, 7,12,17, 5,10,15, 3, 8,13)(18,27,19,28,20,29,21,30,22,31,23,32,24,33,25,34,26)$ |
| 17D4 | $17^{2}$ | $16$ | $17$ | $32$ | $( 1,12, 6,17,11, 5,16,10, 4,15, 9, 3,14, 8, 2,13, 7)(18,33,31,29,27,25,23,21,19,34,32,30,28,26,24,22,20)$ |
| 17D5 | $17^{2}$ | $16$ | $17$ | $32$ | $( 1,16,14,12,10, 8, 6, 4, 2,17,15,13,11, 9, 7, 5, 3)(18,32,29,26,23,20,34,31,28,25,22,19,33,30,27,24,21)$ |
| 17D6 | $17^{2}$ | $16$ | $17$ | $32$ | $( 1,10, 2,11, 3,12, 4,13, 5,14, 6,15, 7,16, 8,17, 9)(18,21,24,27,30,33,19,22,25,28,31,34,20,23,26,29,32)$ |
| 17D7 | $17^{2}$ | $16$ | $17$ | $32$ | $( 1,16,14,12,10, 8, 6, 4, 2,17,15,13,11, 9, 7, 5, 3)(18,23,28,33,21,26,31,19,24,29,34,22,27,32,20,25,30)$ |
| 17D8 | $17^{2}$ | $16$ | $17$ | $32$ | $( 1,16,14,12,10, 8, 6, 4, 2,17,15,13,11, 9, 7, 5, 3)(18,28,21,31,24,34,27,20,30,23,33,26,19,29,22,32,25)$ |
| 34A1 | $34$ | $136$ | $34$ | $33$ | $( 1,23,16,19,14,32,12,28,10,24, 8,20, 6,33, 4,29, 2,25,17,21,15,34,13,30,11,26, 9,22, 7,18, 5,31, 3,27)$ |
| 34A3 | $34$ | $136$ | $34$ | $33$ | $( 1,19,12,24, 6,29,17,34,11,22, 5,27,16,32,10,20, 4,25,15,30, 9,18, 3,23,14,28, 8,33, 2,21,13,26, 7,31)$ |
| 34A5 | $34$ | $136$ | $34$ | $33$ | $( 1,32, 8,29,15,26, 5,23,12,20, 2,34, 9,31,16,28, 6,25,13,22, 3,19,10,33,17,30, 7,27,14,24, 4,21,11,18)$ |
| 34A7 | $34$ | $136$ | $34$ | $33$ | $( 1,28, 4,34, 7,23,10,29,13,18,16,24, 2,30, 5,19, 8,25,11,31,14,20,17,26, 3,32, 6,21, 9,27,12,33,15,22)$ |
| 34A9 | $34$ | $136$ | $34$ | $33$ | $( 1,24,17,22,16,20,15,18,14,33,13,31,12,29,11,27,10,25, 9,23, 8,21, 7,19, 6,34, 5,32, 4,30, 3,28, 2,26)$ |
| 34A11 | $34$ | $136$ | $34$ | $33$ | $( 1,20,13,27, 8,34, 3,24,15,31,10,21, 5,28,17,18,12,25, 7,32, 2,22,14,29, 9,19, 4,26,16,33,11,23, 6,30)$ |
| 34A13 | $34$ | $136$ | $34$ | $33$ | $( 1,33, 9,32,17,31, 8,30,16,29, 7,28,15,27, 6,26,14,25, 5,24,13,23, 4,22,12,21, 3,20,11,19, 2,18,10,34)$ |
| 34A15 | $34$ | $136$ | $34$ | $33$ | $( 1,29, 5,20, 9,28,13,19,17,27, 4,18, 8,26,12,34,16,25, 3,33, 7,24,11,32,15,23, 2,31, 6,22,10,30,14,21)$ |
| 34B1 | $34$ | $136$ | $34$ | $33$ | $( 1,19,15,31,12,26, 9,21, 6,33, 3,28,17,23,14,18,11,30, 8,25, 5,20, 2,32,16,27,13,22,10,34, 7,29, 4,24)$ |
| 34B3 | $34$ | $136$ | $34$ | $33$ | $( 1,31, 9,33,17,18, 8,20,16,22, 7,24,15,26, 6,28,14,30, 5,32,13,34, 4,19,12,21, 3,23,11,25, 2,27,10,29)$ |
| 34B5 | $34$ | $136$ | $34$ | $33$ | $( 1,26, 3,18, 5,27, 7,19, 9,28,11,20,13,29,15,21,17,30, 2,22, 4,31, 6,23, 8,32,10,24,12,33,14,25,16,34)$ |
| 34B7 | $34$ | $136$ | $34$ | $33$ | $( 1,21,14,20,10,19, 6,18, 2,34,15,33,11,32, 7,31, 3,30,16,29,12,28, 8,27, 4,26,17,25,13,24, 9,23, 5,22)$ |
| 34B9 | $34$ | $136$ | $34$ | $33$ | $( 1,33, 8,22,15,28, 5,34,12,23, 2,29, 9,18,16,24, 6,30,13,19, 3,25,10,31,17,20, 7,26,14,32, 4,21,11,27)$ |
| 34B11 | $34$ | $136$ | $34$ | $33$ | $( 1,28, 2,24, 3,20, 4,33, 5,29, 6,25, 7,21, 8,34, 9,30,10,26,11,22,12,18,13,31,14,27,15,23,16,19,17,32)$ |
| 34B13 | $34$ | $136$ | $34$ | $33$ | $( 1,23,13,26, 8,29, 3,32,15,18,10,21, 5,24,17,27,12,30, 7,33, 2,19,14,22, 9,25, 4,28,16,31,11,34, 6,20)$ |
| 34B15 | $34$ | $136$ | $34$ | $33$ | $( 1,18, 7,28,13,21, 2,31, 8,24,14,34, 3,27, 9,20,15,30, 4,23,10,33,16,26, 5,19,11,29,17,22, 6,32,12,25)$ |
Malle's constant $a(G)$: $1/16$
magma: ConjugacyClasses(G);
Character table
49 x 49 character tablemagma: CharacterTable(G);
Regular extensions
Data not computed