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Group invariants
Abstract group: | $F_{17}$ |
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Order: | $272=2^{4} \cdot 17$ |
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Cyclic: | no |
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Abelian: | no |
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Solvable: | yes |
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Nilpotency class: | not nilpotent |
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Group action invariants
Degree $n$: | $34$ |
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Transitive number $t$: | $8$ |
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Parity: | $-1$ |
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Primitive: | no |
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$\card{\Aut(F/K)}$: | $2$ |
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Generators: | $(1,16,20,25,17,6,22,12,14,33,30,23,31,10,27,4)(2,15,19,26,18,5,21,11,13,34,29,24,32,9,28,3)(7,8)$, $(1,31,30,11,20,24,26,9)(2,32,29,12,19,23,25,10)(3,15,22,8,17,5,34,14)(4,16,21,7,18,6,33,13)$ |
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Low degree resolvents
$\card{(G/N)}$ Galois groups for stem field(s) $2$: $C_2$ $4$: $C_4$ $8$: $C_8$ $16$: $C_{16}$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 17: $F_{17}$
Low degree siblings
17T5Siblings 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 | $2^{16},1^{2}$ | $17$ | $2$ | $16$ | $( 1, 9)( 2,10)( 3, 8)( 4, 7)(11,34)(12,33)(13,32)(14,31)(15,30)(16,29)(17,27)(18,28)(19,25)(20,26)(21,23)(22,24)$ |
4A1 | $4^{8},1^{2}$ | $17$ | $4$ | $24$ | $( 1,24, 9,22)( 2,23,10,21)( 3,31, 8,14)( 4,32, 7,13)(11,30,34,15)(12,29,33,16)(17,20,27,26)(18,19,28,25)$ |
4A-1 | $4^{8},1^{2}$ | $17$ | $4$ | $24$ | $( 1,22, 9,24)( 2,21,10,23)( 3,14, 8,31)( 4,13, 7,32)(11,15,34,30)(12,16,33,29)(17,26,27,20)(18,25,28,19)$ |
8A1 | $8^{4},1^{2}$ | $17$ | $8$ | $28$ | $( 1,31,24, 8, 9,14,22, 3)( 2,32,23, 7,10,13,21, 4)(11,17,30,20,34,27,15,26)(12,18,29,19,33,28,16,25)$ |
8A-1 | $8^{4},1^{2}$ | $17$ | $8$ | $28$ | $( 1, 3,22,14, 9, 8,24,31)( 2, 4,21,13,10, 7,23,32)(11,26,15,27,34,20,30,17)(12,25,16,28,33,19,29,18)$ |
8A3 | $8^{4},1^{2}$ | $17$ | $8$ | $28$ | $( 1, 8,22,31, 9, 3,24,14)( 2, 7,21,32,10, 4,23,13)(11,20,15,17,34,26,30,27)(12,19,16,18,33,25,29,28)$ |
8A-3 | $8^{4},1^{2}$ | $17$ | $8$ | $28$ | $( 1,14,24, 3, 9,31,22, 8)( 2,13,23, 4,10,32,21, 7)(11,27,30,26,34,17,15,20)(12,28,29,25,33,18,16,19)$ |
16A1 | $16^{2},2$ | $17$ | $16$ | $31$ | $( 1,29,31,19,24,33, 8,28, 9,16,14,25,22,12, 3,18)( 2,30,32,20,23,34, 7,27,10,15,13,26,21,11, 4,17)( 5, 6)$ |
16A-1 | $16^{2},2$ | $17$ | $16$ | $31$ | $( 1,18, 3,12,22,25,14,16, 9,28, 8,33,24,19,31,29)( 2,17, 4,11,21,26,13,15,10,27, 7,34,23,20,32,30)( 5, 6)$ |
16A3 | $16^{2},2$ | $17$ | $16$ | $31$ | $( 1,19, 8,16,22,18,31,33, 9,25, 3,29,24,28,14,12)( 2,20, 7,15,21,17,32,34,10,26, 4,30,23,27,13,11)( 5, 6)$ |
16A-3 | $16^{2},2$ | $17$ | $16$ | $31$ | $( 1,12,14,28,24,29, 3,25, 9,33,31,18,22,16, 8,19)( 2,11,13,27,23,30, 4,26,10,34,32,17,21,15, 7,20)( 5, 6)$ |
16A5 | $16^{2},2$ | $17$ | $16$ | $31$ | $( 1,33,14,18,24,16, 3,19, 9,12,31,28,22,29, 8,25)( 2,34,13,17,23,15, 4,20,10,11,32,27,21,30, 7,26)( 5, 6)$ |
16A-5 | $16^{2},2$ | $17$ | $16$ | $31$ | $( 1,25, 8,29,22,28,31,12, 9,19, 3,16,24,18,14,33)( 2,26, 7,30,21,27,32,11,10,20, 4,15,23,17,13,34)( 5, 6)$ |
16A7 | $16^{2},2$ | $17$ | $16$ | $31$ | $( 1,28, 3,33,22,19,14,29, 9,18, 8,12,24,25,31,16)( 2,27, 4,34,21,20,13,30,10,17, 7,11,23,26,32,15)( 5, 6)$ |
16A-7 | $16^{2},2$ | $17$ | $16$ | $31$ | $( 1,16,31,25,24,12, 8,18, 9,29,14,19,22,33, 3,28)( 2,15,32,26,23,11, 7,17,10,30,13,20,21,34, 4,27)( 5, 6)$ |
17A | $17^{2}$ | $16$ | $17$ | $32$ | $( 1, 8,14,20,26,31, 3, 9,15,22,27,34, 5,11,17,24,30)( 2, 7,13,19,25,32, 4,10,16,21,28,33, 6,12,18,23,29)$ |
Malle's constant $a(G)$: $1/16$
Character table
1A | 2A | 4A1 | 4A-1 | 8A1 | 8A-1 | 8A3 | 8A-3 | 16A1 | 16A-1 | 16A3 | 16A-3 | 16A5 | 16A-5 | 16A7 | 16A-7 | 17A | ||
Size | 1 | 17 | 17 | 17 | 17 | 17 | 17 | 17 | 17 | 17 | 17 | 17 | 17 | 17 | 17 | 17 | 16 | |
2 P | 1A | 1A | 2A | 2A | 4A1 | 4A-1 | 4A-1 | 4A1 | 8A1 | 8A-1 | 8A3 | 8A-3 | 8A-3 | 8A3 | 8A-1 | 8A1 | 17A | |
17 P | 1A | 2A | 4A1 | 4A-1 | 8A1 | 8A-1 | 8A3 | 8A-3 | 16A1 | 16A-1 | 16A3 | 16A-3 | 16A5 | 16A-5 | 16A7 | 16A-7 | 1A | |
Type | ||||||||||||||||||
272.50.1a | R | |||||||||||||||||
272.50.1b | R | |||||||||||||||||
272.50.1c1 | C | |||||||||||||||||
272.50.1c2 | C | |||||||||||||||||
272.50.1d1 | C | |||||||||||||||||
272.50.1d2 | C | |||||||||||||||||
272.50.1d3 | C | |||||||||||||||||
272.50.1d4 | C | |||||||||||||||||
272.50.1e1 | C | |||||||||||||||||
272.50.1e2 | C | |||||||||||||||||
272.50.1e3 | C | |||||||||||||||||
272.50.1e4 | C | |||||||||||||||||
272.50.1e5 | C | |||||||||||||||||
272.50.1e6 | C | |||||||||||||||||
272.50.1e7 | C | |||||||||||||||||
272.50.1e8 | C | |||||||||||||||||
272.50.16a | R |
Regular extensions
Data not computed