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Group invariants
| Abstract group: | $\SL(2,16).C_4$ |
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| Order: | $16320=2^{6} \cdot 3 \cdot 5 \cdot 17$ |
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| Cyclic: | no |
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| Abelian: | no |
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| Solvable: | no |
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| Nilpotency class: | not nilpotent |
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Group action invariants
| Degree $n$: | $34$ |
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| Transitive number $t$: | $44$ |
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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: | $(1,30,28,16,11,21,24,32)(2,29,27,15,12,22,23,31)(3,34,19,17,26,6,8,9)(4,33,20,18,25,5,7,10)(13,14)$, $(1,30,10,7,24,32,18,34)(2,29,9,8,23,31,17,33)(3,4)(5,21,15,27,11,25,19,13)(6,22,16,28,12,26,20,14)$ |
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Low degree resolvents
$\card{(G/N)}$ Galois groups for stem field(s) $2$: $C_2$ $4$: $C_4$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 17: $\PSL(2,16):C_4$
Low degree siblings
17T8Siblings 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 |
| $1^{34}$ | $1$ | $1$ | $0$ | $()$ | |
| $2^{16},1^{2}$ | $255$ | $2$ | $16$ | $( 1,15)( 2,16)( 3,33)( 4,34)( 5,29)( 6,30)( 9,32)(10,31)(11,18)(12,17)(13,20)(14,19)(21,25)(22,26)(23,27)(24,28)$ | |
| $4^{8},1^{2}$ | $1020$ | $4$ | $24$ | $( 1, 3,15,33)( 2, 4,16,34)( 5,24,29,28)( 6,23,30,27)( 9,13,32,20)(10,14,31,19)(11,22,18,26)(12,21,17,25)$ | |
| $8^{4},2$ | $2040$ | $8$ | $29$ | $( 1,30, 3,27,15, 6,33,23)( 2,29, 4,28,16, 5,34,24)( 7, 8)( 9,22,13,18,32,26,20,11)(10,21,14,17,31,25,19,12)$ | |
| $8^{4},2$ | $2040$ | $8$ | $29$ | $( 1,23,33, 6,15,27, 3,30)( 2,24,34, 5,16,28, 4,29)( 7, 8)( 9,11,20,26,32,18,13,22)(10,12,19,25,31,17,14,21)$ | |
| $2^{12},1^{10}$ | $68$ | $2$ | $12$ | $( 3,14)( 4,13)( 5,15)( 6,16)( 9,21)(10,22)(11,26)(12,25)(17,23)(18,24)(27,32)(28,31)$ | |
| $5^{6},1^{4}$ | $544$ | $5$ | $24$ | $( 1,33, 8,19,29)( 2,34, 7,20,30)( 5,22,31,26,18)( 6,21,32,25,17)( 9,27,12,23,16)(10,28,11,24,15)$ | |
| $10^{2},5^{2},2^{2}$ | $1632$ | $10$ | $28$ | $( 1,19,33,29, 8)( 2,20,34,30, 7)( 3,14)( 4,13)( 5,11,22,24,31,15,26,10,18,28)( 6,12,21,23,32,16,25, 9,17,27)$ | |
| $3^{10},1^{4}$ | $272$ | $3$ | $20$ | $( 1,11,26)( 2,12,25)( 5, 8,15)( 6, 7,16)( 9,21,20)(10,22,19)(17,34,23)(18,33,24)(27,32,30)(28,31,29)$ | |
| $15^{2},1^{4}$ | $1088$ | $15$ | $28$ | $( 1, 5,28,33,22,11, 8,31,24,19,26,15,29,18,10)( 2, 6,27,34,21,12, 7,32,23,20,25,16,30,17, 9)$ | |
| $4^{6},2^{5}$ | $680$ | $4$ | $23$ | $( 1,16,33,13)( 2,15,34,14)( 3,30, 5,20)( 4,29, 6,19)( 7, 8)( 9,22,17,24)(10,21,18,23)(11,32)(12,31)(25,26)(27,28)$ | |
| $4^{6},2^{5}$ | $680$ | $4$ | $23$ | $( 1,13,33,16)( 2,14,34,15)( 3,20, 5,30)( 4,19, 6,29)( 7, 8)( 9,24,17,22)(10,23,18,21)(11,32)(12,31)(25,26)(27,28)$ | |
| $6^{4},3^{2},1^{4}$ | $1360$ | $6$ | $24$ | $( 1,29,10,33,19,18)( 2,30, 9,34,20,17)( 3,24,15, 5,22,14)( 4,23,16, 6,21,13)( 7,27,25)( 8,28,26)$ | |
| $12^{2},6,2^{2}$ | $1360$ | $12$ | $29$ | $( 1,23,29,16,10, 6,33,21,19,13,18, 4)( 2,24,30,15, 9, 5,34,22,20,14,17, 3)( 7,26,27, 8,25,28)(11,32)(12,31)$ | |
| $12^{2},6,2^{2}$ | $1360$ | $12$ | $29$ | $( 1,21,29,13,10, 4,33,23,19,16,18, 6)( 2,22,30,14, 9, 3,34,24,20,15,17, 5)( 7,26,27, 8,25,28)(11,32)(12,31)$ | |
| $17^{2}$ | $960$ | $17$ | $32$ | $( 1, 5, 3,28,26,11,29,19,33,24,18,31,22,15, 8,10,14)( 2, 6, 4,27,25,12,30,20,34,23,17,32,21,16, 7, 9,13)$ | |
| $17^{2}$ | $960$ | $17$ | $32$ | $( 1,29,22, 5,19,15, 3,33, 8,28,24,10,26,18,14,11,31)( 2,30,21, 6,20,16, 4,34, 7,27,23, 9,25,17,13,12,32)$ |
Malle's constant $a(G)$: $1/12$
Character table
Character table not computed
Regular extensions
Data not computed