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Magma
magma: G := TransitiveGroup(34, 7);
Group invariants
Abstract group: | $C_{34}:C_8$ | magma: IdentifyGroup(G);
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Order: | $272=2^{4} \cdot 17$ | 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 $t$: | $7$ | 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: | $(1,34,17,25,21,24,5,31)(2,33,18,26,22,23,6,32)(3,16,10,29,20,8,13,27)(4,15,9,30,19,7,14,28)$, $(1,7,29,19,5,33,11,22)(2,8,30,20,6,34,12,21)(3,4)(9,25,28,24,32,16,14,17)(10,26,27,23,31,15,13,18)$ | 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, $C_2^2$ $8$: $C_8$ x 2, $C_4\times C_2$ $16$: $C_8\times C_2$ $136$: $C_{17}:C_{8}$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 17: $C_{17}:C_{8}$
Low degree siblings
34T7Siblings 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 | $2^{16},1^{2}$ | $17$ | $2$ | $16$ | $( 3,34)( 4,33)( 5,31)( 6,32)( 7,30)( 8,29)( 9,28)(10,27)(11,25)(12,26)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)$ |
2C | $2^{17}$ | $17$ | $2$ | $17$ | $( 1,23)( 2,24)( 3,22)( 4,21)( 5,19)( 6,20)( 7,17)( 8,18)( 9,16)(10,15)(11,14)(12,13)(25,33)(26,34)(27,32)(28,31)(29,30)$ |
4A1 | $4^{8},1^{2}$ | $17$ | $4$ | $24$ | $( 3,10,34,27)( 4, 9,33,28)( 5,17,31,20)( 6,18,32,19)( 7,26,30,12)( 8,25,29,11)(13,16,24,21)(14,15,23,22)$ |
4A-1 | $4^{8},2$ | $17$ | $4$ | $25$ | $( 1,15, 3,23)( 2,16, 4,24)( 5,32,34, 7)( 6,31,33, 8)( 9,13,30,25)(10,14,29,26)(11,22,27,18)(12,21,28,17)(19,20)$ |
4B1 | $4^{8},2$ | $17$ | $4$ | $25$ | $( 1,22,10,23)( 2,21, 9,24)( 3,14, 8,32)( 4,13, 7,31)( 5, 6)(11,15,34,30)(12,16,33,29)(17,26,27,19)(18,25,28,20)$ |
4B-1 | $4^{8},1^{2}$ | $17$ | $4$ | $24$ | $( 3,27,34,10)( 4,28,33, 9)( 5,20,31,17)( 6,19,32,18)( 7,12,30,26)( 8,11,29,25)(13,21,24,16)(14,22,23,15)$ |
8A1 | $8^{4},2$ | $17$ | $8$ | $29$ | $( 1,26, 5,33,21,32,17,23)( 2,25, 6,34,22,31,18,24)( 3,30,13,15,20,28,10, 7)( 4,29,14,16,19,27, 9, 8)(11,12)$ |
8A-1 | $8^{4},2$ | $17$ | $8$ | $29$ | $( 1,12,25,32,20, 9,29,23)( 2,11,26,31,19,10,30,24)( 3, 7,34,15,17,14,21, 6)( 4, 8,33,16,18,13,22, 5)(27,28)$ |
8A3 | $8^{4},1^{2}$ | $17$ | $8$ | $28$ | $( 3,20,27,31,34,17,10, 5)( 4,19,28,32,33,18, 9, 6)( 7,22,12,23,30,15,26,14)( 8,21,11,24,29,16,25,13)$ |
8A-3 | $8^{4},1^{2}$ | $17$ | $8$ | $28$ | $( 3,31,10,20,34, 5,27,17)( 4,32, 9,19,33, 6,28,18)( 7,23,26,22,30,14,12,15)( 8,24,25,21,29,13,11,16)$ |
8B1 | $8^{4},2$ | $17$ | $8$ | $29$ | $( 1, 7,27, 4,25,19,34,23)( 2, 8,28, 3,26,20,33,24)( 5, 9,11,30,21,18,16,32)( 6,10,12,29,22,17,15,31)(13,14)$ |
8B-1 | $8^{4},2$ | $17$ | $8$ | $29$ | $( 1,30,16, 6,27,33,13,23)( 2,29,15, 5,28,34,14,24)( 3,12, 8, 9,25,18,21,19)( 4,11, 7,10,26,17,22,20)(31,32)$ |
8B3 | $8^{4},1^{2}$ | $17$ | $8$ | $28$ | $( 3,17,27, 5,34,20,10,31)( 4,18,28, 6,33,19, 9,32)( 7,15,12,14,30,22,26,23)( 8,16,11,13,29,21,25,24)$ |
8B-3 | $8^{4},1^{2}$ | $17$ | $8$ | $28$ | $( 3, 5,10,17,34,31,27,20)( 4, 6, 9,18,33,32,28,19)( 7,14,26,15,30,23,12,22)( 8,13,25,16,29,24,11,21)$ |
17A1 | $17^{2}$ | $8$ | $17$ | $32$ | $( 1,25,16, 5,29,20,10,34,24,13, 3,27,17, 8,31,21,11)( 2,26,15, 6,30,19, 9,33,23,14, 4,28,18, 7,32,22,12)$ |
17A3 | $17^{2}$ | $8$ | $17$ | $32$ | $( 1, 5,10,13,17,21,25,29,34, 3, 8,11,16,20,24,27,31)( 2, 6, 9,14,18,22,26,30,33, 4, 7,12,15,19,23,28,32)$ |
34A1 | $34$ | $8$ | $34$ | $33$ | $( 1,14,25, 4,16,28, 5,18,29, 7,20,32,10,22,34,12,24, 2,13,26, 3,15,27, 6,17,30, 8,19,31, 9,21,33,11,23)$ |
34A3 | $34$ | $8$ | $34$ | $33$ | $( 1, 4, 5, 7,10,12,13,15,17,19,21,23,25,28,29,32,34, 2, 3, 6, 8, 9,11,14,16,18,20,22,24,26,27,30,31,33)$ |
Malle's constant $a(G)$: $1/16$
magma: ConjugacyClasses(G);
Character table
1A | 2A | 2B | 2C | 4A1 | 4A-1 | 4B1 | 4B-1 | 8A1 | 8A-1 | 8A3 | 8A-3 | 8B1 | 8B-1 | 8B3 | 8B-3 | 17A1 | 17A3 | 34A1 | 34A3 | ||
Size | 1 | 1 | 17 | 17 | 17 | 17 | 17 | 17 | 17 | 17 | 17 | 17 | 17 | 17 | 17 | 17 | 8 | 8 | 8 | 8 | |
2 P | 1A | 1A | 1A | 1A | 2B | 2B | 2B | 2B | 4A1 | 4A1 | 4A-1 | 4A1 | 4A-1 | 4A-1 | 4A-1 | 4A1 | 17A1 | 17A3 | 17A1 | 17A3 | |
17 P | 1A | 2A | 2B | 2C | 4A1 | 4B1 | 4B-1 | 4A-1 | 8A1 | 8A-3 | 8B3 | 8B1 | 8A-1 | 8A3 | 8B-1 | 8B-3 | 1A | 1A | 2A | 2A | |
Type | |||||||||||||||||||||
272.51.1a | R | ||||||||||||||||||||
272.51.1b | R | ||||||||||||||||||||
272.51.1c | R | ||||||||||||||||||||
272.51.1d | R | ||||||||||||||||||||
272.51.1e1 | C | ||||||||||||||||||||
272.51.1e2 | C | ||||||||||||||||||||
272.51.1f1 | C | ||||||||||||||||||||
272.51.1f2 | C | ||||||||||||||||||||
272.51.1g1 | C | ||||||||||||||||||||
272.51.1g2 | C | ||||||||||||||||||||
272.51.1g3 | C | ||||||||||||||||||||
272.51.1g4 | C | ||||||||||||||||||||
272.51.1h1 | C | ||||||||||||||||||||
272.51.1h2 | C | ||||||||||||||||||||
272.51.1h3 | C | ||||||||||||||||||||
272.51.1h4 | C | ||||||||||||||||||||
272.51.8a1 | R | ||||||||||||||||||||
272.51.8a2 | R | ||||||||||||||||||||
272.51.8b1 | R | ||||||||||||||||||||
272.51.8b2 | R |
magma: CharacterTable(G);
Regular extensions
Data not computed