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Magma
magma: G := TransitiveGroup(34, 28);
Group invariants
Abstract group: | $\SOMinus(4,4)$ | magma: IdentifyGroup(G);
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Order: | $8160=2^{5} \cdot 3 \cdot 5 \cdot 17$ | 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: | no | 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 $t$: | $28$ | 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: | $(1,5,29,2,6,30)(3,23,33,15,20,9)(4,24,34,16,19,10)(7,21,25,13,32,17)(8,22,26,14,31,18)(11,12)(27,28)$, $(1,25,18,32,7,16,29,14,10,6,3,33,22,20,24,28,12)(2,26,17,31,8,15,30,13,9,5,4,34,21,19,23,27,11)$ | magma: Generators(G);
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Low degree resolvents
$\card{(G/N)}$ Galois groups for stem field(s) $2$: $C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 17: $\PSL(2,16):C_2$
Low degree siblings
17T7Siblings 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 | $2^{17}$ | $68$ | $2$ | $17$ | $( 1,26)( 2,25)( 3,13)( 4,14)( 5, 7)( 6, 8)( 9,10)(11,12)(15,16)(17,33)(18,34)(19,22)(20,21)(23,24)(27,28)(29,31)(30,32)$ |
2B | $2^{16},1^{2}$ | $255$ | $2$ | $16$ | $( 1,14)( 2,13)( 3,25)( 4,26)( 5, 8)( 6, 7)( 9,27)(10,28)(15,23)(16,24)(17,31)(18,32)(19,21)(20,22)(29,33)(30,34)$ |
3A | $3^{10},1^{4}$ | $272$ | $3$ | $20$ | $( 1,24,33)( 2,23,34)( 3,16,32)( 4,15,31)( 5,19, 8)( 6,20, 7)( 9,30,26)(10,29,25)(13,27,17)(14,28,18)$ |
4A | $4^{8},2$ | $1020$ | $4$ | $25$ | $( 1,34,14,30)( 2,33,13,29)( 3,17,25,31)( 4,18,26,32)( 5,24, 8,16)( 6,23, 7,15)( 9,20,27,22)(10,19,28,21)(11,12)$ |
5A1 | $5^{6},1^{4}$ | $272$ | $5$ | $24$ | $( 1, 7, 3,10,28)( 2, 8, 4, 9,27)( 5,15,30,17,23)( 6,16,29,18,24)(13,34,19,31,26)(14,33,20,32,25)$ |
5A2 | $5^{6},1^{4}$ | $272$ | $5$ | $24$ | $( 1, 3,28, 7,10)( 2, 4,27, 8, 9)( 5,30,23,15,17)( 6,29,24,16,18)(13,19,26,34,31)(14,20,25,33,32)$ |
6A | $6^{5},2^{2}$ | $1360$ | $6$ | $27$ | $( 1, 8,22,26, 6,19)( 2, 7,21,25, 5,20)( 3,17,29,13,33,31)( 4,18,30,14,34,32)( 9,10)(11,28,23,12,27,24)(15,16)$ |
10A1 | $10^{3},2^{2}$ | $816$ | $10$ | $29$ | $( 1, 8,25,13,22,34,20, 9,28,11)( 2, 7,26,14,21,33,19,10,27,12)( 3,31)( 4,32)( 5,18,23,29,15, 6,17,24,30,16)$ |
10A3 | $10^{3},2^{2}$ | $816$ | $10$ | $29$ | $( 1,13,20,11,25,34,28, 8,22, 9)( 2,14,19,12,26,33,27, 7,21,10)( 3,31)( 4,32)( 5,29,17,16,23, 6,30,18,15,24)$ |
15A1 | $15^{2},1^{4}$ | $544$ | $15$ | $28$ | $( 1,32,18, 7,25,24, 3,14, 6,10,33,16,28,20,29)( 2,31,17, 8,26,23, 4,13, 5, 9,34,15,27,19,30)$ |
15A2 | $15^{2},1^{4}$ | $544$ | $15$ | $28$ | $( 1,18,25, 3, 6,33,28,29,32, 7,24,14,10,16,20)( 2,17,26, 4, 5,34,27,30,31, 8,23,13, 9,15,19)$ |
17A1 | $17^{2}$ | $480$ | $17$ | $32$ | $( 1,24,14,16, 6,28,29,25, 7,20,12,33,18,10,22, 3,32)( 2,23,13,15, 5,27,30,26, 8,19,11,34,17, 9,21, 4,31)$ |
17A2 | $17^{2}$ | $480$ | $17$ | $32$ | $( 1,14, 6,29, 7,12,18,22,32,24,16,28,25,20,33,10, 3)( 2,13, 5,30, 8,11,17,21,31,23,15,27,26,19,34, 9, 4)$ |
17A3 | $17^{2}$ | $480$ | $17$ | $32$ | $( 1,16,29,20,18, 3,24, 6,25,12,10,32,14,28, 7,33,22)( 2,15,30,19,17, 4,23, 5,26,11, 9,31,13,27, 8,34,21)$ |
17A6 | $17^{2}$ | $480$ | $17$ | $32$ | $( 1,29,18,24,25,10,14, 7,22,16,20, 3, 6,12,32,28,33)( 2,30,17,23,26, 9,13, 8,21,15,19, 4, 5,11,31,27,34)$ |
Malle's constant $a(G)$: $1/16$
magma: ConjugacyClasses(G);
Character table
1A | 2A | 2B | 3A | 4A | 5A1 | 5A2 | 6A | 10A1 | 10A3 | 15A1 | 15A2 | 17A1 | 17A2 | 17A3 | 17A6 | ||
Size | 1 | 68 | 255 | 272 | 1020 | 272 | 272 | 1360 | 816 | 816 | 544 | 544 | 480 | 480 | 480 | 480 | |
2 P | 1A | 1A | 1A | 3A | 2B | 5A2 | 5A1 | 3A | 5A1 | 5A2 | 15A2 | 15A1 | 17A2 | 17A1 | 17A6 | 17A3 | |
3 P | 1A | 2A | 2B | 1A | 4A | 5A2 | 5A1 | 2A | 10A3 | 10A1 | 5A1 | 5A2 | 17A3 | 17A6 | 17A2 | 17A1 | |
5 P | 1A | 2A | 2B | 3A | 4A | 1A | 1A | 6A | 2A | 2A | 3A | 3A | 17A3 | 17A6 | 17A2 | 17A1 | |
17 P | 1A | 2A | 2B | 3A | 4A | 5A2 | 5A1 | 6A | 10A3 | 10A1 | 15A2 | 15A1 | 1A | 1A | 1A | 1A | |
Type | |||||||||||||||||
8160.a.1a | R | ||||||||||||||||
8160.a.1b | R | ||||||||||||||||
8160.a.16a | R | ||||||||||||||||
8160.a.16b | R | ||||||||||||||||
8160.a.17a | R | ||||||||||||||||
8160.a.17b | R | ||||||||||||||||
8160.a.17c1 | R | ||||||||||||||||
8160.a.17c2 | R | ||||||||||||||||
8160.a.17d1 | R | ||||||||||||||||
8160.a.17d2 | R | ||||||||||||||||
8160.a.30a1 | R | ||||||||||||||||
8160.a.30a2 | R | ||||||||||||||||
8160.a.30a3 | R | ||||||||||||||||
8160.a.30a4 | R | ||||||||||||||||
8160.a.34a1 | R | ||||||||||||||||
8160.a.34a2 | R |
magma: CharacterTable(G);
Regular extensions
Data not computed