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Magma
magma: G := TransitiveGroup(34, 5);
Group invariants
Abstract group: | $C_{34}:C_4$ | 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 $t$: | $5$ | 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: | $(1,9,8,33)(2,10,7,34)(3,17,6,26)(4,18,5,25)(11,16,31,27)(12,15,32,28)(13,23,29,19)(14,24,30,20)$, $(1,26,20,29)(2,25,19,30)(3,33,17,21)(4,34,18,22)(5,7,16,13)(6,8,15,14)(9,23,11,32)(10,24,12,31)(27,28)$ | 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, $C_2^2$ $8$: $C_4\times C_2$ $68$: $C_{17}:C_{4}$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 17: $C_{17}:C_{4}$
Low degree siblings
34T5Siblings 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 | $2^{16},1^{2}$ | $17$ | $2$ | $16$ | $( 3,34)( 4,33)( 5,31)( 6,32)( 7,29)( 8,30)( 9,27)(10,28)(11,25)(12,26)(13,23)(14,24)(15,22)(16,21)(17,19)(18,20)$ |
2C | $2^{17}$ | $17$ | $2$ | $17$ | $( 1, 6)( 2, 5)( 3, 4)( 7,33)( 8,34)( 9,32)(10,31)(11,29)(12,30)(13,27)(14,28)(15,25)(16,26)(17,24)(18,23)(19,21)(20,22)$ |
4A1 | $4^{8},1^{2}$ | $17$ | $4$ | $24$ | $( 3,10,34,28)( 4, 9,33,27)( 5,18,31,20)( 6,17,32,19)( 7,26,29,12)( 8,25,30,11)(13,15,23,22)(14,16,24,21)$ |
4A-1 | $4^{8},1^{2}$ | $17$ | $4$ | $24$ | $( 3,28,34,10)( 4,27,33, 9)( 5,20,31,18)( 6,19,32,17)( 7,12,29,26)( 8,11,30,25)(13,22,23,15)(14,21,24,16)$ |
4B1 | $4^{8},2$ | $17$ | $4$ | $25$ | $( 1,19,24, 6)( 2,20,23, 5)( 3,27,22,31)( 4,28,21,32)( 7, 9,17,16)( 8,10,18,15)(11,26,14,34)(12,25,13,33)(29,30)$ |
4B-1 | $4^{8},2$ | $17$ | $4$ | $25$ | $( 1,17,21, 6)( 2,18,22, 5)( 3, 9,19,14)( 4,10,20,13)( 7,27,15,30)( 8,28,16,29)(11,12)(23,31,34,25)(24,32,33,26)$ |
17A1 | $17^{2}$ | $4$ | $17$ | $32$ | $( 1,11,21,31, 8,18,27, 4,14,24,33, 9,20,30, 5,16,25)( 2,12,22,32, 7,17,28, 3,13,23,34,10,19,29, 6,15,26)$ |
17A2 | $17^{2}$ | $4$ | $17$ | $32$ | $( 1,21, 8,27,14,33,20, 5,25,11,31,18, 4,24, 9,30,16)( 2,22, 7,28,13,34,19, 6,26,12,32,17, 3,23,10,29,15)$ |
17A3 | $17^{2}$ | $4$ | $17$ | $32$ | $( 1,27,20,11, 4,30,21,14, 5,31,24,16, 8,33,25,18, 9)( 2,28,19,12, 3,29,22,13, 6,32,23,15, 7,34,26,17,10)$ |
17A6 | $17^{2}$ | $4$ | $17$ | $32$ | $( 1,20, 4,21, 5,24, 8,25, 9,27,11,30,14,31,16,33,18)( 2,19, 3,22, 6,23, 7,26,10,28,12,29,13,32,15,34,17)$ |
34A1 | $34$ | $4$ | $34$ | $33$ | $( 1, 7,14,19,25,32, 4,10,16,22,27,34, 5,12,18,23,30, 2, 8,13,20,26,31, 3, 9,15,21,28,33, 6,11,17,24,29)$ |
34A3 | $34$ | $4$ | $34$ | $33$ | $( 1,34,31,29,27,26,24,22,20,17,16,13,11,10, 8, 6, 4, 2,33,32,30,28,25,23,21,19,18,15,14,12, 9, 7, 5, 3)$ |
34A7 | $34$ | $4$ | $34$ | $33$ | $( 1,32,27,23,20,15,11, 7, 4,34,30,26,21,17,14,10, 5, 2,31,28,24,19,16,12, 8, 3,33,29,25,22,18,13, 9, 6)$ |
34A9 | $34$ | $4$ | $34$ | $33$ | $( 1,23,11,34,21,10,31,19, 8,29,18, 6,27,15, 4,26,14, 2,24,12,33,22, 9,32,20, 7,30,17, 5,28,16, 3,25,13)$ |
Malle's constant $a(G)$: $1/16$
magma: ConjugacyClasses(G);
Character table
1A | 2A | 2B | 2C | 4A1 | 4A-1 | 4B1 | 4B-1 | 17A1 | 17A2 | 17A3 | 17A6 | 34A1 | 34A3 | 34A7 | 34A9 | ||
Size | 1 | 1 | 17 | 17 | 17 | 17 | 17 | 17 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | |
2 P | 1A | 1A | 1A | 1A | 2B | 2B | 2B | 2B | 17A6 | 17A3 | 17A2 | 17A1 | 17A6 | 17A2 | 17A1 | 17A3 | |
17 P | 1A | 2A | 2B | 2C | 4A1 | 4A-1 | 4B1 | 4B-1 | 1A | 1A | 1A | 1A | 2A | 2A | 2A | 2A | |
Type | |||||||||||||||||
136.13.1a | R | ||||||||||||||||
136.13.1b | R | ||||||||||||||||
136.13.1c | R | ||||||||||||||||
136.13.1d | R | ||||||||||||||||
136.13.1e1 | C | ||||||||||||||||
136.13.1e2 | C | ||||||||||||||||
136.13.1f1 | C | ||||||||||||||||
136.13.1f2 | C | ||||||||||||||||
136.13.4a1 | R | ||||||||||||||||
136.13.4a2 | R | ||||||||||||||||
136.13.4a3 | R | ||||||||||||||||
136.13.4a4 | R | ||||||||||||||||
136.13.4b1 | R | ||||||||||||||||
136.13.4b2 | R | ||||||||||||||||
136.13.4b3 | R | ||||||||||||||||
136.13.4b4 | R |
magma: CharacterTable(G);
Regular extensions
Data not computed