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Magma
magma: G := TransitiveGroup(34, 14);
Group invariants
Abstract group: | $C_{17}^2:Q_8$ | magma: IdentifyGroup(G);
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Order: | $2312=2^{3} \cdot 17^{2}$ | 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 $t$: | $14$ | 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: | $(1,25,16,18)(2,20,15,23)(3,32,14,28)(4,27,13,33)(5,22,12,21)(6,34,11,26)(7,29,10,31)(8,24,9,19)(17,30)$, $(1,17,13,14)(2,4,12,10)(3,8,11,6)(5,16,9,15)(18,20,29,27)(19,33,28,31)(21,25,26,22)(23,34,24,30)$ | magma: Generators(G);
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Low degree resolvents
$\card{(G/N)}$ Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_2^2$ $8$: $Q_8$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 17: None
Low degree siblings
34T14 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 | $2^{16},1^{2}$ | $289$ | $2$ | $16$ | $( 1, 2)( 3,17)( 4,16)( 5,15)( 6,14)( 7,13)( 8,12)( 9,11)(18,19)(20,34)(21,33)(22,32)(23,31)(24,30)(25,29)(26,28)$ |
4A | $4^{8},2$ | $578$ | $4$ | $25$ | $( 1,33, 2,21)( 3,26,17,28)( 4,31,16,23)( 5,19,15,18)( 6,24,14,30)( 7,29,13,25)( 8,34,12,20)( 9,22,11,32)(10,27)$ |
4B | $4^{8},2$ | $578$ | $4$ | $25$ | $( 1,24,15,33)( 2,21,14,19)( 3,18,13,22)( 4,32,12,25)( 5,29,11,28)( 6,26,10,31)( 7,23, 9,34)( 8,20)(16,30,17,27)$ |
4C | $4^{8},1^{2}$ | $578$ | $4$ | $24$ | $( 1, 8,14, 7)( 2, 4,13,11)( 3,17,12,15)( 5, 9,10, 6)(19,22,34,31)(20,26,33,27)(21,30,32,23)(24,25,29,28)$ |
17A1 | $17,1^{17}$ | $8$ | $17$ | $16$ | $(18,25,32,22,29,19,26,33,23,30,20,27,34,24,31,21,28)$ |
17A2 | $17,1^{17}$ | $8$ | $17$ | $16$ | $(18,32,29,26,23,20,34,31,28,25,22,19,33,30,27,24,21)$ |
17A3 | $17,1^{17}$ | $8$ | $17$ | $16$ | $(18,22,26,30,34,21,25,29,33,20,24,28,32,19,23,27,31)$ |
17A6 | $17,1^{17}$ | $8$ | $17$ | $16$ | $(18,26,34,25,33,24,32,23,31,22,30,21,29,20,28,19,27)$ |
17B1 | $17^{2}$ | $8$ | $17$ | $32$ | $( 1, 8,15, 5,12, 2, 9,16, 6,13, 3,10,17, 7,14, 4,11)(18,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19)$ |
17B2 | $17^{2}$ | $8$ | $17$ | $32$ | $( 1,15,12, 9, 6, 3,17,14,11, 8, 5, 2,16,13,10, 7, 4)(18,33,31,29,27,25,23,21,19,34,32,30,28,26,24,22,20)$ |
17B3 | $17^{2}$ | $8$ | $17$ | $32$ | $( 1, 5, 9,13,17, 4, 8,12,16, 3, 7,11,15, 2, 6,10,14)(18,32,29,26,23,20,34,31,28,25,22,19,33,30,27,24,21)$ |
17B6 | $17^{2}$ | $8$ | $17$ | $32$ | $( 1, 9,17, 8,16, 7,15, 6,14, 5,13, 4,12, 3,11, 2,10)(18,29,23,34,28,22,33,27,21,32,26,20,31,25,19,30,24)$ |
17C1 | $17^{2}$ | $8$ | $17$ | $32$ | $( 1, 8,15, 5,12, 2, 9,16, 6,13, 3,10,17, 7,14, 4,11)(18,31,27,23,19,32,28,24,20,33,29,25,21,34,30,26,22)$ |
17C2 | $17^{2}$ | $8$ | $17$ | $32$ | $( 1,15,12, 9, 6, 3,17,14,11, 8, 5, 2,16,13,10, 7, 4)(18,27,19,28,20,29,21,30,22,31,23,32,24,33,25,34,26)$ |
17C3 | $17^{2}$ | $8$ | $17$ | $32$ | $( 1, 5, 9,13,17, 4, 8,12,16, 3, 7,11,15, 2, 6,10,14)(18,23,28,33,21,26,31,19,24,29,34,22,27,32,20,25,30)$ |
17C6 | $17^{2}$ | $8$ | $17$ | $32$ | $( 1,16,14,12,10, 8, 6, 4, 2,17,15,13,11, 9, 7, 5, 3)(18,29,23,34,28,22,33,27,21,32,26,20,31,25,19,30,24)$ |
17D1 | $17^{2}$ | $8$ | $17$ | $32$ | $( 1, 8,15, 5,12, 2, 9,16, 6,13, 3,10,17, 7,14, 4,11)(18,24,30,19,25,31,20,26,32,21,27,33,22,28,34,23,29)$ |
17D2 | $17^{2}$ | $8$ | $17$ | $32$ | $( 1,15,12, 9, 6, 3,17,14,11, 8, 5, 2,16,13,10, 7, 4)(18,30,25,20,32,27,22,34,29,24,19,31,26,21,33,28,23)$ |
17D3 | $17^{2}$ | $8$ | $17$ | $32$ | $( 1, 5, 9,13,17, 4, 8,12,16, 3, 7,11,15, 2, 6,10,14)(18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34)$ |
17D4 | $17^{2}$ | $8$ | $17$ | $32$ | $( 1, 9,17, 8,16, 7,15, 6,14, 5,13, 4,12, 3,11, 2,10)(18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34)$ |
17D5 | $17^{2}$ | $8$ | $17$ | $32$ | $( 1, 8,15, 5,12, 2, 9,16, 6,13, 3,10,17, 7,14, 4,11)(18,21,24,27,30,33,19,22,25,28,31,34,20,23,26,29,32)$ |
17D6 | $17^{2}$ | $8$ | $17$ | $32$ | $( 1, 9,17, 8,16, 7,15, 6,14, 5,13, 4,12, 3,11, 2,10)(18,20,22,24,26,28,30,32,34,19,21,23,25,27,29,31,33)$ |
17D7 | $17^{2}$ | $8$ | $17$ | $32$ | $( 1,15,12, 9, 6, 3,17,14,11, 8, 5, 2,16,13,10, 7, 4)(18,24,30,19,25,31,20,26,32,21,27,33,22,28,34,23,29)$ |
17D8 | $17^{2}$ | $8$ | $17$ | $32$ | $( 1,15,12, 9, 6, 3,17,14,11, 8, 5, 2,16,13,10, 7, 4)(18,23,28,33,21,26,31,19,24,29,34,22,27,32,20,25,30)$ |
17E1 | $17^{2}$ | $8$ | $17$ | $32$ | $( 1, 5, 9,13,17, 4, 8,12,16, 3, 7,11,15, 2, 6,10,14)(18,22,26,30,34,21,25,29,33,20,24,28,32,19,23,27,31)$ |
17E2 | $17^{2}$ | $8$ | $17$ | $32$ | $( 1, 9,17, 8,16, 7,15, 6,14, 5,13, 4,12, 3,11, 2,10)(18,26,34,25,33,24,32,23,31,22,30,21,29,20,28,19,27)$ |
17E3 | $17^{2}$ | $8$ | $17$ | $32$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17)(18,27,19,28,20,29,21,30,22,31,23,32,24,33,25,34,26)$ |
17E4 | $17^{2}$ | $8$ | $17$ | $32$ | $( 1,12, 6,17,11, 5,16,10, 4,15, 9, 3,14, 8, 2,13, 7)(18,21,24,27,30,33,19,22,25,28,31,34,20,23,26,29,32)$ |
17E5 | $17^{2}$ | $8$ | $17$ | $32$ | $( 1, 5, 9,13,17, 4, 8,12,16, 3, 7,11,15, 2, 6,10,14)(18,20,22,24,26,28,30,32,34,19,21,23,25,27,29,31,33)$ |
17E6 | $17^{2}$ | $8$ | $17$ | $32$ | $( 1, 8,15, 5,12, 2, 9,16, 6,13, 3,10,17, 7,14, 4,11)(18,25,32,22,29,19,26,33,23,30,20,27,34,24,31,21,28)$ |
17E7 | $17^{2}$ | $8$ | $17$ | $32$ | $( 1, 8,15, 5,12, 2, 9,16, 6,13, 3,10,17, 7,14, 4,11)(18,28,21,31,24,34,27,20,30,23,33,26,19,29,22,32,25)$ |
17E8 | $17^{2}$ | $8$ | $17$ | $32$ | $( 1, 6,11,16, 4, 9,14, 2, 7,12,17, 5,10,15, 3, 8,13)(18,24,30,19,25,31,20,26,32,21,27,33,22,28,34,23,29)$ |
17F1 | $17^{2}$ | $8$ | $17$ | $32$ | $( 1, 5, 9,13,17, 4, 8,12,16, 3, 7,11,15, 2, 6,10,14)(18,29,23,34,28,22,33,27,21,32,26,20,31,25,19,30,24)$ |
17F2 | $17^{2}$ | $8$ | $17$ | $32$ | $( 1, 9,17, 8,16, 7,15, 6,14, 5,13, 4,12, 3,11, 2,10)(18,23,28,33,21,26,31,19,24,29,34,22,27,32,20,25,30)$ |
17F3 | $17^{2}$ | $8$ | $17$ | $32$ | $( 1,12, 6,17,11, 5,16,10, 4,15, 9, 3,14, 8, 2,13, 7)(18,33,31,29,27,25,23,21,19,34,32,30,28,26,24,22,20)$ |
17F4 | $17^{2}$ | $8$ | $17$ | $32$ | $( 1,10, 2,11, 3,12, 4,13, 5,14, 6,15, 7,16, 8,17, 9)(18,21,24,27,30,33,19,22,25,28,31,34,20,23,26,29,32)$ |
17F5 | $17^{2}$ | $8$ | $17$ | $32$ | $( 1,15,12, 9, 6, 3,17,14,11, 8, 5, 2,16,13,10, 7, 4)(18,31,27,23,19,32,28,24,20,33,29,25,21,34,30,26,22)$ |
17F6 | $17^{2}$ | $8$ | $17$ | $32$ | $( 1, 6,11,16, 4, 9,14, 2, 7,12,17, 5,10,15, 3, 8,13)(18,31,27,23,19,32,28,24,20,33,29,25,21,34,30,26,22)$ |
17F7 | $17^{2}$ | $8$ | $17$ | $32$ | $( 1,15,12, 9, 6, 3,17,14,11, 8, 5, 2,16,13,10, 7, 4)(18,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19)$ |
17F8 | $17^{2}$ | $8$ | $17$ | $32$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17)(18,24,30,19,25,31,20,26,32,21,27,33,22,28,34,23,29)$ |
Malle's constant $a(G)$: $1/16$
magma: ConjugacyClasses(G);
Character table
41 x 41 character tablemagma: CharacterTable(G);
Regular extensions
Data not computed