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Magma
magma: G := TransitiveGroup(34, 24);
Group invariants
| Abstract group: | $D_{17}^2.C_2^2$ | 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 $t$: | $24$ | 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: | $(1,23,17,28)(2,18,16,33)(3,30,15,21)(4,25,14,26)(5,20,13,31)(6,32,12,19)(7,27,11,24)(8,22,10,29)(9,34)$, $(1,9,11,3)(2,5,10,7)(4,14,8,15)(12,16,17,13)(18,23,20,32)(21,28,34,27)(22,24,33,31)(25,29,30,26)$, $(1,7,13,2,8,14,3,9,15,4,10,16,5,11,17,6,12)(18,24)(19,23)(20,22)(25,34)(26,33)(27,32)(28,31)(29,30)$ | magma: Generators(G);
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Low degree resolvents
$\card{(G/N)}$ Galois groups for stem field(s) $2$: $C_2$ x 7 $4$: $C_2^2$ x 7 $8$: $C_2^3$ $16$: $Q_8:C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 17: None
Low degree siblings
34T24 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 | $2^{17}$ | $34$ | $2$ | $17$ | $( 1,28)( 2,25)( 3,22)( 4,19)( 5,33)( 6,30)( 7,27)( 8,24)( 9,21)(10,18)(11,32)(12,29)(13,26)(14,23)(15,20)(16,34)(17,31)$ |
| 2C | $2^{17}$ | $34$ | $2$ | $17$ | $( 1,31)( 2,19)( 3,24)( 4,29)( 5,34)( 6,22)( 7,27)( 8,32)( 9,20)(10,25)(11,30)(12,18)(13,23)(14,28)(15,33)(16,21)(17,26)$ |
| 2D | $2^{16},1^{2}$ | $289$ | $2$ | $16$ | $( 1, 2)( 3,17)( 4,16)( 5,15)( 6,14)( 7,13)( 8,12)( 9,11)(18,32)(19,31)(20,30)(21,29)(22,28)(23,27)(24,26)(33,34)$ |
| 4A1 | $4^{8},1^{2}$ | $289$ | $4$ | $24$ | $( 1,14,15, 2)( 3, 5,13,11)( 4, 9,12, 7)( 6,17,10,16)(18,26,24,33)(19,30,23,29)(20,34,22,25)(27,28,32,31)$ |
| 4A-1 | $4^{8},1^{2}$ | $289$ | $4$ | $24$ | $( 1, 2,15,14)( 3,11,13, 5)( 4, 7,12, 9)( 6,16,10,17)(18,33,24,26)(19,29,23,30)(20,25,22,34)(27,31,32,28)$ |
| 4B | $4^{8},2$ | $578$ | $4$ | $25$ | $( 1,19, 2,31)( 3,26,17,24)( 4,21,16,29)( 5,33,15,34)( 6,28,14,22)( 7,23,13,27)( 8,18,12,32)( 9,30,11,20)(10,25)$ |
| 4C | $4^{8},1^{2}$ | $578$ | $4$ | $24$ | $( 1,16, 8,10)( 2, 3, 7, 6)( 4,11, 5,15)( 9,14,17,12)(18,28,22,29)(19,24,21,33)(23,25,34,32)(26,30,31,27)$ |
| 4D | $4^{8},2$ | $578$ | $4$ | $25$ | $( 1,33)( 2,30,17,19)( 3,27,16,22)( 4,24,15,25)( 5,21,14,28)( 6,18,13,31)( 7,32,12,34)( 8,29,11,20)( 9,26,10,23)$ |
| 17A1 | $17,1^{17}$ | $8$ | $17$ | $16$ | $(18,30,25,20,32,27,22,34,29,24,19,31,26,21,33,28,23)$ |
| 17A2 | $17,1^{17}$ | $8$ | $17$ | $16$ | $(18,25,32,22,29,19,26,33,23,30,20,27,34,24,31,21,28)$ |
| 17A3 | $17,1^{17}$ | $8$ | $17$ | $16$ | $(18,20,22,24,26,28,30,32,34,19,21,23,25,27,29,31,33)$ |
| 17A6 | $17,1^{17}$ | $8$ | $17$ | $16$ | $(18,22,26,30,34,21,25,29,33,20,24,28,32,19,23,27,31)$ |
| 17B1 | $17^{2}$ | $8$ | $17$ | $32$ | $( 1,17,16,15,14,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(18,21,24,27,30,33,19,22,25,28,31,34,20,23,26,29,32)$ |
| 17B2 | $17^{2}$ | $8$ | $17$ | $32$ | $( 1,16,14,12,10, 8, 6, 4, 2,17,15,13,11, 9, 7, 5, 3)(18,24,30,19,25,31,20,26,32,21,27,33,22,28,34,23,29)$ |
| 17B3 | $17^{2}$ | $8$ | $17$ | $32$ | $( 1,15,12, 9, 6, 3,17,14,11, 8, 5, 2,16,13,10, 7, 4)(18,27,19,28,20,29,21,30,22,31,23,32,24,33,25,34,26)$ |
| 17B6 | $17^{2}$ | $8$ | $17$ | $32$ | $( 1,12, 6,17,11, 5,16,10, 4,15, 9, 3,14, 8, 2,13, 7)(18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34)$ |
| 17C1 | $17^{2}$ | $8$ | $17$ | $32$ | $( 1,15,12, 9, 6, 3,17,14,11, 8, 5, 2,16,13,10, 7, 4)(18,20,22,24,26,28,30,32,34,19,21,23,25,27,29,31,33)$ |
| 17C2 | $17^{2}$ | $8$ | $17$ | $32$ | $( 1,12, 6,17,11, 5,16,10, 4,15, 9, 3,14, 8, 2,13, 7)(18,22,26,30,34,21,25,29,33,20,24,28,32,19,23,27,31)$ |
| 17C3 | $17^{2}$ | $8$ | $17$ | $32$ | $( 1, 9,17, 8,16, 7,15, 6,14, 5,13, 4,12, 3,11, 2,10)(18,24,30,19,25,31,20,26,32,21,27,33,22,28,34,23,29)$ |
| 17C6 | $17^{2}$ | $8$ | $17$ | $32$ | $( 1,17,16,15,14,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(18,30,25,20,32,27,22,34,29,24,19,31,26,21,33,28,23)$ |
| 17D1 | $17^{2}$ | $16$ | $17$ | $32$ | $( 1,14,10, 6, 2,15,11, 7, 3,16,12, 8, 4,17,13, 9, 5)(18,22,26,30,34,21,25,29,33,20,24,28,32,19,23,27,31)$ |
| 17D2 | $17^{2}$ | $16$ | $17$ | $32$ | $( 1,10, 2,11, 3,12, 4,13, 5,14, 6,15, 7,16, 8,17, 9)(18,26,34,25,33,24,32,23,31,22,30,21,29,20,28,19,27)$ |
| 17D3 | $17^{2}$ | $16$ | $17$ | $32$ | $( 1,17,16,15,14,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(18,27,19,28,20,29,21,30,22,31,23,32,24,33,25,34,26)$ |
| 17D6 | $17^{2}$ | $16$ | $17$ | $32$ | $( 1, 9,17, 8,16, 7,15, 6,14, 5,13, 4,12, 3,11, 2,10)(18,31,27,23,19,32,28,24,20,33,29,25,21,34,30,26,22)$ |
| 17E1 | $17^{2}$ | $16$ | $17$ | $32$ | $( 1,15,12, 9, 6, 3,17,14,11, 8, 5, 2,16,13,10, 7, 4)(18,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19)$ |
| 17E2 | $17^{2}$ | $16$ | $17$ | $32$ | $( 1,12, 6,17,11, 5,16,10, 4,15, 9, 3,14, 8, 2,13, 7)(18,33,31,29,27,25,23,21,19,34,32,30,28,26,24,22,20)$ |
| 17E3 | $17^{2}$ | $16$ | $17$ | $32$ | $( 1, 9,17, 8,16, 7,15, 6,14, 5,13, 4,12, 3,11, 2,10)(18,32,29,26,23,20,34,31,28,25,22,19,33,30,27,24,21)$ |
| 17E6 | $17^{2}$ | $16$ | $17$ | $32$ | $( 1,16,14,12,10, 8, 6, 4, 2,17,15,13,11, 9, 7, 5, 3)(18,32,29,26,23,20,34,31,28,25,22,19,33,30,27,24,21)$ |
| 17F1 | $17^{2}$ | $16$ | $17$ | $32$ | $( 1, 4, 7,10,13,16, 2, 5, 8,11,14,17, 3, 6, 9,12,15)(18,23,28,33,21,26,31,19,24,29,34,22,27,32,20,25,30)$ |
| 17F2 | $17^{2}$ | $16$ | $17$ | $32$ | $( 1,11, 4,14, 7,17,10, 3,13, 6,16, 9, 2,12, 5,15, 8)(18,24,30,19,25,31,20,26,32,21,27,33,22,28,34,23,29)$ |
| 17F3 | $17^{2}$ | $16$ | $17$ | $32$ | $( 1,13, 8, 3,15,10, 5,17,12, 7, 2,14, 9, 4,16,11, 6)(18,28,21,31,24,34,27,20,30,23,33,26,19,29,22,32,25)$ |
| 17F6 | $17^{2}$ | $16$ | $17$ | $32$ | $( 1,11, 4,14, 7,17,10, 3,13, 6,16, 9, 2,12, 5,15, 8)(18,21,24,27,30,33,19,22,25,28,31,34,20,23,26,29,32)$ |
| 34A1 | $17,2^{8},1$ | $136$ | $34$ | $24$ | $( 1,10)( 2, 9)( 3, 8)( 4, 7)( 5, 6)(11,17)(12,16)(13,15)(18,24,30,19,25,31,20,26,32,21,27,33,22,28,34,23,29)$ |
| 34A3 | $17,2^{8},1$ | $136$ | $34$ | $24$ | $( 1,10)( 2, 9)( 3, 8)( 4, 7)( 5, 6)(11,17)(12,16)(13,15)(18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34)$ |
| 34A7 | $17,2^{8},1$ | $136$ | $34$ | $24$ | $( 1,10)( 2, 9)( 3, 8)( 4, 7)( 5, 6)(11,17)(12,16)(13,15)(18,26,34,25,33,24,32,23,31,22,30,21,29,20,28,19,27)$ |
| 34A9 | $17,2^{8},1$ | $136$ | $34$ | $24$ | $( 1,10)( 2, 9)( 3, 8)( 4, 7)( 5, 6)(11,17)(12,16)(13,15)(18,21,24,27,30,33,19,22,25,28,31,34,20,23,26,29,32)$ |
| 34B1 | $34$ | $136$ | $34$ | $33$ | $( 1,21,17,24,16,27,15,30,14,33,13,19,12,22,11,25,10,28, 9,31, 8,34, 7,20, 6,23, 5,26, 4,29, 3,32, 2,18)$ |
| 34B3 | $34$ | $136$ | $34$ | $33$ | $( 1,24,15,33,12,25, 9,34, 6,26, 3,18,17,27,14,19,11,28, 8,20, 5,29, 2,21,16,30,13,22,10,31, 7,23, 4,32)$ |
| 34B7 | $34$ | $136$ | $34$ | $33$ | $( 1,30,11,34, 4,21,14,25, 7,29,17,33,10,20, 3,24,13,28, 6,32,16,19, 9,23, 2,27,12,31, 5,18,15,22, 8,26)$ |
| 34B9 | $34$ | $136$ | $34$ | $33$ | $( 1,33, 9,26,17,19, 8,29,16,22, 7,32,15,25, 6,18,14,28, 5,21,13,31, 4,24,12,34, 3,27,11,20, 2,30,10,23)$ |
| 34C1 | $34$ | $136$ | $34$ | $33$ | $( 1,32,15,34,12,19, 9,21, 6,23, 3,25,17,27,14,29,11,31, 8,33, 5,18, 2,20,16,22,13,24,10,26, 7,28, 4,30)$ |
| 34C3 | $34$ | $136$ | $34$ | $33$ | $( 1,34, 9,23,17,29, 8,18,16,24, 7,30,15,19, 6,25,14,31, 5,20,13,26, 4,32,12,21, 3,27,11,33, 2,22,10,28)$ |
| 34C7 | $34$ | $136$ | $34$ | $33$ | $( 1,21,14,18,10,32, 6,29, 2,26,15,23,11,20, 7,34, 3,31,16,28,12,25, 8,22, 4,19,17,33,13,30, 9,27, 5,24)$ |
| 34C9 | $34$ | $136$ | $34$ | $33$ | $( 1,23, 8,24,15,25, 5,26,12,27, 2,28, 9,29,16,30, 6,31,13,32, 3,33,10,34,17,18, 7,19,14,20, 4,21,11,22)$ |
Malle's constant $a(G)$: $1/8$
magma: ConjugacyClasses(G);
Character table
46 x 46 character tablemagma: CharacterTable(G);
Regular extensions
Data not computed