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Magma
magma: G := TransitiveGroup(34, 3);
Group invariants
Abstract group: | $D_{34}$ | magma: IdentifyGroup(G);
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Order: | $68=2^{2} \cdot 17$ | 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 $t$: | $3$ | 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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$\card{\Aut(F/K)}$: | $2$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | $(1,5,10,14,17,22,25,29,33,3,8,11,16,20,23,27,32,2,6,9,13,18,21,26,30,34,4,7,12,15,19,24,28,31)$, $(1,24)(2,23)(3,21)(4,22)(5,19)(6,20)(7,17)(8,18)(9,16)(10,15)(11,13)(12,14)(25,34)(26,33)(27,32)(28,31)(29,30)$ | magma: Generators(G);
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Low degree resolvents
$\card{(G/N)}$ Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_2^2$ $34$: $D_{17}$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 17: $D_{17}$
Low degree siblings
34T3Siblings 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 | $2^{17}$ | $1$ | $2$ | $17$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)$ |
2B | $2^{16},1^{2}$ | $17$ | $2$ | $16$ | $( 3,34)( 4,33)( 5,31)( 6,32)( 7,29)( 8,30)( 9,27)(10,28)(11,26)(12,25)(13,23)(14,24)(15,22)(16,21)(17,19)(18,20)$ |
2C | $2^{17}$ | $17$ | $2$ | $17$ | $( 1,31)( 2,32)( 3,30)( 4,29)( 5,28)( 6,27)( 7,25)( 8,26)( 9,23)(10,24)(11,21)(12,22)(13,20)(14,19)(15,17)(16,18)(33,34)$ |
17A1 | $17^{2}$ | $2$ | $17$ | $32$ | $( 1,28,19,12, 4,30,21,13, 6,32,23,16, 8,33,25,17,10)( 2,27,20,11, 3,29,22,14, 5,31,24,15, 7,34,26,18, 9)$ |
17A2 | $17^{2}$ | $2$ | $17$ | $32$ | $( 1,19, 4,21, 6,23, 8,25,10,28,12,30,13,32,16,33,17)( 2,20, 3,22, 5,24, 7,26, 9,27,11,29,14,31,15,34,18)$ |
17A3 | $17^{2}$ | $2$ | $17$ | $32$ | $( 1,12,21,32, 8,17,28, 4,13,23,33,10,19,30, 6,16,25)( 2,11,22,31, 7,18,27, 3,14,24,34, 9,20,29, 5,15,26)$ |
17A4 | $17^{2}$ | $2$ | $17$ | $32$ | $( 1, 4, 6, 8,10,12,13,16,17,19,21,23,25,28,30,32,33)( 2, 3, 5, 7, 9,11,14,15,18,20,22,24,26,27,29,31,34)$ |
17A5 | $17^{2}$ | $2$ | $17$ | $32$ | $( 1,30,23,17,12, 6,33,28,21,16,10, 4,32,25,19,13, 8)( 2,29,24,18,11, 5,34,27,22,15, 9, 3,31,26,20,14, 7)$ |
17A6 | $17^{2}$ | $2$ | $17$ | $32$ | $( 1,21, 8,28,13,33,19, 6,25,12,32,17, 4,23,10,30,16)( 2,22, 7,27,14,34,20, 5,26,11,31,18, 3,24, 9,29,15)$ |
17A7 | $17^{2}$ | $2$ | $17$ | $32$ | $( 1,13,25, 4,16,28, 6,17,30, 8,19,32,10,21,33,12,23)( 2,14,26, 3,15,27, 5,18,29, 7,20,31, 9,22,34,11,24)$ |
17A8 | $17^{2}$ | $2$ | $17$ | $32$ | $( 1, 6,10,13,17,21,25,30,33, 4, 8,12,16,19,23,28,32)( 2, 5, 9,14,18,22,26,29,34, 3, 7,11,15,20,24,27,31)$ |
34A1 | $34$ | $2$ | $34$ | $33$ | $( 1,31,28,24,19,15,12, 7, 4,34,30,26,21,18,13, 9, 6, 2,32,27,23,20,16,11, 8, 3,33,29,25,22,17,14,10, 5)$ |
34A3 | $34$ | $2$ | $34$ | $33$ | $( 1,24,12,34,21, 9,32,20, 8,29,17, 5,28,15, 4,26,13, 2,23,11,33,22,10,31,19, 7,30,18, 6,27,16, 3,25,14)$ |
34A5 | $34$ | $2$ | $34$ | $33$ | $( 1,15,30, 9,23, 3,17,31,12,26, 6,20,33,14,28, 7,21, 2,16,29,10,24, 4,18,32,11,25, 5,19,34,13,27, 8,22)$ |
34A7 | $34$ | $2$ | $34$ | $33$ | $( 1, 7,13,20,25,31, 4, 9,16,22,28,34, 6,11,17,24,30, 2, 8,14,19,26,32, 3,10,15,21,27,33, 5,12,18,23,29)$ |
34A9 | $34$ | $2$ | $34$ | $33$ | $( 1,34,32,29,28,26,23,22,19,18,16,14,12, 9, 8, 5, 4, 2,33,31,30,27,25,24,21,20,17,15,13,11,10, 7, 6, 3)$ |
34A11 | $34$ | $2$ | $34$ | $33$ | $( 1,26,16, 5,30,20,10,34,23,14, 4,27,17, 7,32,22,12, 2,25,15, 6,29,19, 9,33,24,13, 3,28,18, 8,31,21,11)$ |
34A13 | $34$ | $2$ | $34$ | $33$ | $( 1,18,33,15,32,14,30,11,28, 9,25, 7,23, 5,21, 3,19, 2,17,34,16,31,13,29,12,27,10,26, 8,24, 6,22, 4,20)$ |
34A15 | $34$ | $2$ | $34$ | $33$ | $( 1, 9,17,26,33, 7,16,24,32, 5,13,22,30, 3,12,20,28, 2,10,18,25,34, 8,15,23,31, 6,14,21,29, 4,11,19,27)$ |
Malle's constant $a(G)$: $1/16$
magma: ConjugacyClasses(G);
Character table
1A | 2A | 2B | 2C | 17A1 | 17A2 | 17A3 | 17A4 | 17A5 | 17A6 | 17A7 | 17A8 | 34A1 | 34A3 | 34A5 | 34A7 | 34A9 | 34A11 | 34A13 | 34A15 | ||
Size | 1 | 1 | 17 | 17 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | |
2 P | 1A | 1A | 1A | 1A | 17A2 | 17A4 | 17A6 | 17A8 | 17A7 | 17A5 | 17A3 | 17A1 | 17A1 | 17A3 | 17A5 | 17A7 | 17A8 | 17A6 | 17A4 | 17A2 | |
17 P | 1A | 2A | 2B | 2C | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 2A | 2A | 2A | 2A | 2A | 2A | 2A | 2A | |
Type | |||||||||||||||||||||
68.4.1a | R | ||||||||||||||||||||
68.4.1b | R | ||||||||||||||||||||
68.4.1c | R | ||||||||||||||||||||
68.4.1d | R | ||||||||||||||||||||
68.4.2a1 | R | ||||||||||||||||||||
68.4.2a2 | R | ||||||||||||||||||||
68.4.2a3 | R | ||||||||||||||||||||
68.4.2a4 | R | ||||||||||||||||||||
68.4.2a5 | R | ||||||||||||||||||||
68.4.2a6 | R | ||||||||||||||||||||
68.4.2a7 | R | ||||||||||||||||||||
68.4.2a8 | R | ||||||||||||||||||||
68.4.2b1 | R | ||||||||||||||||||||
68.4.2b2 | R | ||||||||||||||||||||
68.4.2b3 | R | ||||||||||||||||||||
68.4.2b4 | R | ||||||||||||||||||||
68.4.2b5 | R | ||||||||||||||||||||
68.4.2b6 | R | ||||||||||||||||||||
68.4.2b7 | R | ||||||||||||||||||||
68.4.2b8 | R |
magma: CharacterTable(G);
Regular extensions
Data not computed