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Magma
magma: G := TransitiveGroup(34, 26);
Group invariants
Abstract group: | $C_{17}^2:\OD_{16}$ | 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 $t$: | $26$ | 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: | $(1,12,7,17,14,3,8,15)(2,10,11,9,13,5,4,6)(18,21,27,22,29,26,20,25)(19,23,31,30,28,24,33,34)$, $(1,34,5,28,4,21,17,27)(2,24,9,22,3,31,13,33)(6,18,8,32,16,20,14,23)(7,25,12,26,15,30,10,29)(11,19)$ | magma: Generators(G);
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Low degree resolvents
$\card{(G/N)}$ Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_4$ x 2, $C_2^2$ $8$: $C_4\times C_2$ $16$: $C_8:C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 17: None
Low degree siblings
34T25, 34T26Siblings 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 | $2^{17}$ | $34$ | $2$ | $17$ | $( 1,22)( 2,25)( 3,28)( 4,31)( 5,34)( 6,20)( 7,23)( 8,26)( 9,29)(10,32)(11,18)(12,21)(13,24)(14,27)(15,30)(16,33)(17,19)$ |
2B | $2^{16},1^{2}$ | $289$ | $2$ | $16$ | $( 1,16)( 2,15)( 3,14)( 4,13)( 5,12)( 6,11)( 7,10)( 8, 9)(18,29)(19,28)(20,27)(21,26)(22,25)(23,24)(30,34)(31,33)$ |
4A1 | $4^{8},1^{2}$ | $289$ | $4$ | $24$ | $( 1, 4,16,13)( 2, 8,15, 9)( 3,12,14, 5)( 6, 7,11,10)(18,27,29,20)(19,31,28,33)(21,22,26,25)(23,30,24,34)$ |
4A-1 | $4^{8},1^{2}$ | $289$ | $4$ | $24$ | $( 1,13,16, 4)( 2, 9,15, 8)( 3, 5,14,12)( 6,10,11, 7)(18,20,29,27)(19,33,28,31)(21,25,26,22)(23,34,24,30)$ |
4B | $4^{8},2$ | $578$ | $4$ | $25$ | $( 1,33,14,19)( 2,28,13,24)( 3,23,12,29)( 4,18,11,34)( 5,30,10,22)( 6,25, 9,27)( 7,20, 8,32)(15,31,17,21)(16,26)$ |
8A1 | $8^{4},2$ | $578$ | $8$ | $29$ | $( 1,25, 4,21,16,22,13,26)( 2,18, 8,27,15,29, 9,20)( 3,28,12,33,14,19, 5,31)( 6,24, 7,34,11,23,10,30)(17,32)$ |
8A-1 | $8^{4},2$ | $578$ | $8$ | $29$ | $( 1,26,13,22,16,21, 4,25)( 2,20, 9,29,15,27, 8,18)( 3,31, 5,19,14,33,12,28)( 6,30,10,23,11,34, 7,24)(17,32)$ |
8B1 | $8^{4},1^{2}$ | $578$ | $8$ | $28$ | $( 1, 5,13,12,10, 6,15,16)( 2, 7,17, 3, 9, 4,11, 8)(18,26,27,25,29,21,20,22)(19,24,31,34,28,23,33,30)$ |
8B-1 | $8^{4},1^{2}$ | $578$ | $8$ | $28$ | $( 1,16,15, 6,10,12,13, 5)( 2, 8,11, 4, 9, 3,17, 7)(18,22,20,21,29,25,27,26)(19,30,33,23,28,34,31,24)$ |
17A1 | $17^{2}$ | $8$ | $17$ | $32$ | $( 1,16,14,12,10, 8, 6, 4, 2,17,15,13,11, 9, 7, 5, 3)(18,29,23,34,28,22,33,27,21,32,26,20,31,25,19,30,24)$ |
17A2 | $17^{2}$ | $8$ | $17$ | $32$ | $( 1,14,10, 6, 2,15,11, 7, 3,16,12, 8, 4,17,13, 9, 5)(18,23,28,33,21,26,31,19,24,29,34,22,27,32,20,25,30)$ |
17A3 | $17^{2}$ | $8$ | $17$ | $32$ | $( 1,12, 6,17,11, 5,16,10, 4,15, 9, 3,14, 8, 2,13, 7)(18,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19)$ |
17A6 | $17^{2}$ | $8$ | $17$ | $32$ | $( 1, 6,11,16, 4, 9,14, 2, 7,12,17, 5,10,15, 3, 8,13)(18,33,31,29,27,25,23,21,19,34,32,30,28,26,24,22,20)$ |
17B1 | $17,1^{17}$ | $16$ | $17$ | $16$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17)$ |
17B3 | $17,1^{17}$ | $16$ | $17$ | $16$ | $( 1, 4, 7,10,13,16, 2, 5, 8,11,14,17, 3, 6, 9,12,15)$ |
17C1 | $17^{2}$ | $16$ | $17$ | $32$ | $( 1, 5, 9,13,17, 4, 8,12,16, 3, 7,11,15, 2, 6,10,14)(18,32,29,26,23,20,34,31,28,25,22,19,33,30,27,24,21)$ |
17C3 | $17^{2}$ | $16$ | $17$ | $32$ | $( 1, 8,15, 5,12, 2, 9,16, 6,13, 3,10,17, 7,14, 4,11)(18,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19)$ |
17D1 | $17^{2}$ | $16$ | $17$ | $32$ | $( 1,11, 4,14, 7,17,10, 3,13, 6,16, 9, 2,12, 5,15, 8)(18,25,32,22,29,19,26,33,23,30,20,27,34,24,31,21,28)$ |
17D2 | $17^{2}$ | $16$ | $17$ | $32$ | $( 1, 4, 7,10,13,16, 2, 5, 8,11,14,17, 3, 6, 9,12,15)(18,32,29,26,23,20,34,31,28,25,22,19,33,30,27,24,21)$ |
17D3 | $17^{2}$ | $16$ | $17$ | $32$ | $( 1,14,10, 6, 2,15,11, 7, 3,16,12, 8, 4,17,13, 9, 5)(18,22,26,30,34,21,25,29,33,20,24,28,32,19,23,27,31)$ |
17D6 | $17^{2}$ | $16$ | $17$ | $32$ | $( 1,10, 2,11, 3,12, 4,13, 5,14, 6,15, 7,16, 8,17, 9)(18,26,34,25,33,24,32,23,31,22,30,21,29,20,28,19,27)$ |
17E1 | $17^{2}$ | $16$ | $17$ | $32$ | $( 1,12, 6,17,11, 5,16,10, 4,15, 9, 3,14, 8, 2,13, 7)(18,25,32,22,29,19,26,33,23,30,20,27,34,24,31,21,28)$ |
17E2 | $17^{2}$ | $16$ | $17$ | $32$ | $( 1, 8,15, 5,12, 2, 9,16, 6,13, 3,10,17, 7,14, 4,11)(18,29,23,34,28,22,33,27,21,32,26,20,31,25,19,30,24)$ |
17E3 | $17^{2}$ | $16$ | $17$ | $32$ | $( 1,16,14,12,10, 8, 6, 4, 2,17,15,13,11, 9, 7, 5, 3)(18,27,19,28,20,29,21,30,22,31,23,32,24,33,25,34,26)$ |
17E6 | $17^{2}$ | $16$ | $17$ | $32$ | $( 1,13, 8, 3,15,10, 5,17,12, 7, 2,14, 9, 4,16,11, 6)(18,25,32,22,29,19,26,33,23,30,20,27,34,24,31,21,28)$ |
17F1 | $17^{2}$ | $16$ | $17$ | $32$ | $( 1,15,12, 9, 6, 3,17,14,11, 8, 5, 2,16,13,10, 7, 4)(18,22,26,30,34,21,25,29,33,20,24,28,32,19,23,27,31)$ |
17F2 | $17^{2}$ | $16$ | $17$ | $32$ | $( 1,11, 4,14, 7,17,10, 3,13, 6,16, 9, 2,12, 5,15, 8)(18,26,34,25,33,24,32,23,31,22,30,21,29,20,28,19,27)$ |
17F3 | $17^{2}$ | $16$ | $17$ | $32$ | $( 1, 3, 5, 7, 9,11,13,15,17, 2, 4, 6, 8,10,12,14,16)(18,21,24,27,30,33,19,22,25,28,31,34,20,23,26,29,32)$ |
17F6 | $17^{2}$ | $16$ | $17$ | $32$ | $( 1,17,16,15,14,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(18,24,30,19,25,31,20,26,32,21,27,33,22,28,34,23,29)$ |
34A1 | $34$ | $136$ | $34$ | $33$ | $( 1,19,16,30,14,24,12,18,10,29, 8,23, 6,34, 4,28, 2,22,17,33,15,27,13,21,11,32, 9,26, 7,20, 5,31, 3,25)$ |
34A3 | $34$ | $136$ | $34$ | $33$ | $( 1,30,12,29, 6,28,17,27,11,26, 5,25,16,24,10,23, 4,22,15,21, 9,20, 3,19,14,18, 8,34, 2,33,13,32, 7,31)$ |
34A7 | $34$ | $136$ | $34$ | $33$ | $( 1,18, 4,27, 7,19,10,28,13,20,16,29, 2,21, 5,30, 8,22,11,31,14,23,17,32, 3,24, 6,33, 9,25,12,34,15,26)$ |
34A9 | $34$ | $136$ | $34$ | $33$ | $( 1,29,17,26,16,23,15,20,14,34,13,31,12,28,11,25,10,22, 9,19, 8,33, 7,30, 6,27, 5,24, 4,21, 3,18, 2,32)$ |
Malle's constant $a(G)$: $1/16$
magma: ConjugacyClasses(G);
Character table
34 x 34 character tablemagma: CharacterTable(G);
Regular extensions
Data not computed