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Magma
magma: G := TransitiveGroup(34, 6);
Group invariants
Abstract group: | $C_{17}:C_8$ | magma: IdentifyGroup(G);
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Order: | $136=2^{3} \cdot 17$ | 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 $t$: | $6$ | 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: | $(1,28,24,21,4,12,15,17)(2,27,23,22,3,11,16,18)(5,30,7,14,34,9,31,25)(6,29,8,13,33,10,32,26)(19,20)$, $(1,20)(2,19)(3,17)(4,18)(5,16)(6,15)(7,13)(8,14)(9,11)(10,12)(21,34)(22,33)(23,31)(24,32)(25,30)(26,29)$ | magma: Generators(G);
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Low degree resolvents
$\card{(G/N)}$ Galois groups for stem field(s) $2$: $C_2$ $4$: $C_4$ $8$: $C_8$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 17: $C_{17}:C_{8}$
Low degree siblings
17T4Siblings 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 | $2^{16},1^{2}$ | $17$ | $2$ | $16$ | $( 1, 4)( 2, 3)( 5,34)( 6,33)( 7,31)( 8,32)( 9,30)(10,29)(11,27)(12,28)(13,26)(14,25)(15,24)(16,23)(17,21)(18,22)$ |
4A1 | $4^{8},1^{2}$ | $17$ | $4$ | $24$ | $( 1,24, 4,15)( 2,23, 3,16)( 5, 7,34,31)( 6, 8,33,32)( 9,25,30,14)(10,26,29,13)(11,18,27,22)(12,17,28,21)$ |
4A-1 | $4^{8},1^{2}$ | $17$ | $4$ | $24$ | $( 1,15, 4,24)( 2,16, 3,23)( 5,31,34, 7)( 6,32,33, 8)( 9,14,30,25)(10,13,29,26)(11,22,27,18)(12,21,28,17)$ |
8A1 | $8^{4},2$ | $17$ | $8$ | $29$ | $( 1,28,24,21, 4,12,15,17)( 2,27,23,22, 3,11,16,18)( 5,30, 7,14,34, 9,31,25)( 6,29, 8,13,33,10,32,26)(19,20)$ |
8A-1 | $8^{4},2$ | $17$ | $8$ | $29$ | $( 1,17,15,12, 4,21,24,28)( 2,18,16,11, 3,22,23,27)( 5,25,31, 9,34,14, 7,30)( 6,26,32,10,33,13, 8,29)(19,20)$ |
8A3 | $8^{4},2$ | $17$ | $8$ | $29$ | $( 1,21,15,28, 4,17,24,12)( 2,22,16,27, 3,18,23,11)( 5,14,31,30,34,25, 7, 9)( 6,13,32,29,33,26, 8,10)(19,20)$ |
8A-3 | $8^{4},2$ | $17$ | $8$ | $29$ | $( 1,12,24,17, 4,28,15,21)( 2,11,23,18, 3,27,16,22)( 5, 9, 7,25,34,30,31,14)( 6,10, 8,26,33,29,32,13)(19,20)$ |
17A1 | $17^{2}$ | $8$ | $17$ | $32$ | $( 1,32,27,24,20,15,11, 8, 4,33,30,25,22,18,14, 9, 6)( 2,31,28,23,19,16,12, 7, 3,34,29,26,21,17,13,10, 5)$ |
17A3 | $17^{2}$ | $8$ | $17$ | $32$ | $( 1,24,11,33,22, 9,32,20, 8,30,18, 6,27,15, 4,25,14)( 2,23,12,34,21,10,31,19, 7,29,17, 5,28,16, 3,26,13)$ |
Malle's constant $a(G)$: $1/16$
magma: ConjugacyClasses(G);
Character table
1A | 2A | 4A1 | 4A-1 | 8A1 | 8A-1 | 8A3 | 8A-3 | 17A1 | 17A3 | ||
Size | 1 | 17 | 17 | 17 | 17 | 17 | 17 | 17 | 8 | 8 | |
2 P | 1A | 1A | 2A | 2A | 4A1 | 4A-1 | 4A-1 | 4A1 | 17A1 | 17A3 | |
17 P | 1A | 2A | 4A1 | 4A-1 | 8A1 | 8A-1 | 8A3 | 8A-3 | 1A | 1A | |
Type | |||||||||||
136.12.1a | R | ||||||||||
136.12.1b | R | ||||||||||
136.12.1c1 | C | ||||||||||
136.12.1c2 | C | ||||||||||
136.12.1d1 | C | ||||||||||
136.12.1d2 | C | ||||||||||
136.12.1d3 | C | ||||||||||
136.12.1d4 | C | ||||||||||
136.12.8a1 | R | ||||||||||
136.12.8a2 | R |
magma: CharacterTable(G);
Regular extensions
Data not computed