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Magma
magma: G := TransitiveGroup(34, 48);
Group invariants
| Abstract group: | $D_{17}^2.D_8$ | magma: IdentifyGroup(G);
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| Order: | $18496=2^{6} \cdot 17^{2}$ | 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 $t$: | $48$ | 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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| $\card{\Aut(F/K)}$: | $1$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | $(1,25,7,31)(2,26,6,30)(3,27,5,29)(4,28)(8,32,17,24)(9,33,16,23)(10,34,15,22)(11,18,14,21)(12,19,13,20)$, $(1,10,4,8,11,9,16,17,5,13,2,15,12,14,7,6)(18,27,22,21,31,33,30,26,32,23,28,29,19,34,20,24)$, $(1,30,7,31,13,32,2,33,8,34,14,18,3,19,9,20,15,21,4,22,10,23,16,24,5,25,11,26,17,27,6,28,12,29)$ | magma: Generators(G);
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Low degree resolvents
$\card{(G/N)}$ Galois groups for stem field(s) $2$: $C_2$ x 7 $4$: $C_2^2$ x 7 $8$: $D_{4}$ x 2, $C_2^3$ $16$: $D_{8}$ x 2, $D_4\times C_2$ $32$: 16T29 $64$: 32T303 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 17: None
Low degree siblings
There are no siblings 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 | $2^{8},1^{18}$ | $34$ | $2$ | $8$ | $( 1,12)( 2,11)( 3,10)( 4, 9)( 5, 8)( 6, 7)(13,17)(14,16)$ |
| 2B | $2^{17}$ | $136$ | $2$ | $17$ | $( 1,20)( 2,21)( 3,22)( 4,23)( 5,24)( 6,25)( 7,26)( 8,27)( 9,28)(10,29)(11,30)(12,31)(13,32)(14,33)(15,34)(16,18)(17,19)$ |
| 2C | $2^{17}$ | $136$ | $2$ | $17$ | $( 1,26)( 2,31)( 3,19)( 4,24)( 5,29)( 6,34)( 7,22)( 8,27)( 9,32)(10,20)(11,25)(12,30)(13,18)(14,23)(15,28)(16,33)(17,21)$ |
| 2D | $2^{16},1^{2}$ | $289$ | $2$ | $16$ | $( 1,12)( 2,11)( 3,10)( 4, 9)( 5, 8)( 6, 7)(13,17)(14,16)(18,33)(19,32)(20,31)(21,30)(22,29)(23,28)(24,27)(25,26)$ |
| 4A1 | $4^{8},1^{2}$ | $289$ | $4$ | $24$ | $( 1, 5, 6, 2)( 3,14, 4,10)( 7,15,17, 9)( 8,11,16,13)(18,24,34,28)(19,20,33,32)(21,29,31,23)(22,25,30,27)$ |
| 4A-1 | $4^{8},1^{2}$ | $289$ | $4$ | $24$ | $( 1, 2, 6, 5)( 3,10, 4,14)( 7, 9,17,15)( 8,13,16,11)(18,28,34,24)(19,32,33,20)(21,23,31,29)(22,27,30,25)$ |
| 4B | $4^{8},1^{2}$ | $578$ | $4$ | $24$ | $( 1,10,12, 3)( 2,14,11,16)( 4, 5, 9, 8)( 6,13, 7,17)(18,30,33,21)(19,26,32,25)(20,22,31,29)(23,27,28,24)$ |
| 4C | $4^{8},2$ | $2312$ | $4$ | $25$ | $( 1,25, 5,33)( 2,27, 4,31)( 3,29)( 6,18,17,23)( 7,20,16,21)( 8,22,15,19)( 9,24,14,34)(10,26,13,32)(11,28,12,30)$ |
| 4D | $4^{8},2$ | $2312$ | $4$ | $25$ | $( 1,28,10,34)( 2,23, 9,22)( 3,18, 8,27)( 4,30, 7,32)( 5,25, 6,20)(11,29,17,33)(12,24,16,21)(13,19,15,26)(14,31)$ |
| 8A1 | $8^{4},1^{2}$ | $578$ | $8$ | $28$ | $( 1, 9,10, 8,12, 4, 3, 5)( 2, 7,14,17,11, 6,16,13)(18,25,30,19,33,26,21,32)(20,24,22,23,31,27,29,28)$ |
| 8A3 | $8^{4},1^{2}$ | $578$ | $8$ | $28$ | $( 1, 8, 3, 9,12, 5,10, 4)( 2,17,16, 7,11,13,14, 6)(18,19,21,25,33,32,30,26)(20,23,29,24,31,28,22,27)$ |
| 8B1 | $8^{4},1^{2}$ | $578$ | $8$ | $28$ | $( 1, 6,13,16,10, 5,15,12)( 2, 4,17, 8, 9, 7,11, 3)(18,34,25,29,31,32,24,20)(19,26,21,27,30,23,28,22)$ |
| 8B-1 | $8^{4},1^{2}$ | $578$ | $8$ | $28$ | $( 1,12,15, 5,10,16,13, 6)( 2, 3,11, 7, 9, 8,17, 4)(18,20,24,32,31,29,25,34)(19,22,28,23,30,27,21,26)$ |
| 16A1 | $16^{2},1^{2}$ | $578$ | $16$ | $30$ | $( 1,11, 9, 6,10,16, 8,13,12, 2, 4, 7, 3,14, 5,17)(18,29,25,28,30,20,19,24,33,22,26,23,21,31,32,27)$ |
| 16A3 | $16^{2},1^{2}$ | $578$ | $16$ | $30$ | $( 1, 6, 8, 2, 3,17, 9,16,12, 7, 5,11,10,13, 4,14)(18,28,19,22,21,27,25,20,33,23,32,29,30,24,26,31)$ |
| 16A5 | $16^{2},1^{2}$ | $578$ | $16$ | $30$ | $( 1,16, 4,17,10, 2, 5, 6,12,14, 9,13, 3,11, 8, 7)(18,20,26,27,30,22,32,28,33,31,25,24,21,29,19,23)$ |
| 16A7 | $16^{2},1^{2}$ | $578$ | $16$ | $30$ | $( 1,13, 5,16, 3, 6, 4,11,12,17, 8,14,10, 7, 9, 2)(18,24,32,20,21,28,26,29,33,27,19,31,30,23,25,22)$ |
| 16B1 | $16^{2},1^{2}$ | $578$ | $16$ | $30$ | $( 1,17, 6, 4,16,12, 2,11, 8, 9, 3, 5,10,14, 7,15)(18,31,19,34,28,27,24,32,22,26,21,23,29,30,33,25)$ |
| 16B-1 | $16^{2},1^{2}$ | $578$ | $16$ | $30$ | $( 1,15, 7,14,10, 5, 3, 9, 8,11, 2,12,16, 4, 6,17)(18,25,33,30,29,23,21,26,22,32,24,27,28,34,19,31)$ |
| 16B3 | $16^{2},1^{2}$ | $578$ | $16$ | $30$ | $( 1, 4, 2, 9,10,15, 6,12, 8, 5, 7,17,16,11, 3,14)(18,34,24,26,29,25,19,27,22,23,33,31,28,32,21,30)$ |
| 16B-3 | $16^{2},1^{2}$ | $578$ | $16$ | $30$ | $( 1,14, 3,11,16,17, 7, 5, 8,12, 6,15,10, 9, 2, 4)(18,30,21,32,28,31,33,23,22,27,19,25,29,26,24,34)$ |
| 17A | $17,1^{17}$ | $32$ | $17$ | $16$ | $(18,31,27,23,19,32,28,24,20,33,29,25,21,34,30,26,22)$ |
| 17B1 | $17^{2}$ | $32$ | $17$ | $32$ | $( 1,12, 6,17,11, 5,16,10, 4,15, 9, 3,14, 8, 2,13, 7)(18,29,23,34,28,22,33,27,21,32,26,20,31,25,19,30,24)$ |
| 17B2 | $17^{2}$ | $32$ | $17$ | $32$ | $( 1, 6,11,16, 4, 9,14, 2, 7,12,17, 5,10,15, 3, 8,13)(18,23,28,33,21,26,31,19,24,29,34,22,27,32,20,25,30)$ |
| 17B3 | $17^{2}$ | $32$ | $17$ | $32$ | $( 1,17,16,15,14,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(18,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19)$ |
| 17B6 | $17^{2}$ | $32$ | $17$ | $32$ | $( 1,16,14,12,10, 8, 6, 4, 2,17,15,13,11, 9, 7, 5, 3)(18,33,31,29,27,25,23,21,19,34,32,30,28,26,24,22,20)$ |
| 17C1 | $17^{2}$ | $32$ | $17$ | $32$ | $( 1, 3, 5, 7, 9,11,13,15,17, 2, 4, 6, 8,10,12,14,16)(18,28,21,31,24,34,27,20,30,23,33,26,19,29,22,32,25)$ |
| 17C2 | $17^{2}$ | $32$ | $17$ | $32$ | $( 1, 5, 9,13,17, 4, 8,12,16, 3, 7,11,15, 2, 6,10,14)(18,21,24,27,30,33,19,22,25,28,31,34,20,23,26,29,32)$ |
| 17C3 | $17^{2}$ | $32$ | $17$ | $32$ | $( 1, 7,13, 2, 8,14, 3, 9,15, 4,10,16, 5,11,17, 6,12)(18,31,27,23,19,32,28,24,20,33,29,25,21,34,30,26,22)$ |
| 17C6 | $17^{2}$ | $32$ | $17$ | $32$ | $( 1,13, 8, 3,15,10, 5,17,12, 7, 2,14, 9, 4,16,11, 6)(18,27,19,28,20,29,21,30,22,31,23,32,24,33,25,34,26)$ |
| 34A | $17,2^{8},1$ | $544$ | $34$ | $24$ | $( 1,12)( 2,11)( 3,10)( 4, 9)( 5, 8)( 6, 7)(13,17)(14,16)(18,33,31,29,27,25,23,21,19,34,32,30,28,26,24,22,20)$ |
| 34B1 | $34$ | $544$ | $34$ | $33$ | $( 1,34,12,28, 6,22,17,33,11,27, 5,21,16,32,10,26, 4,20,15,31, 9,25, 3,19,14,30, 8,24, 2,18,13,29, 7,23)$ |
| 34B3 | $34$ | $544$ | $34$ | $33$ | $( 1,28,17,27,16,26,15,25,14,24,13,23,12,22,11,21,10,20, 9,19, 8,18, 7,34, 6,33, 5,32, 4,31, 3,30, 2,29)$ |
| 34B7 | $34$ | $544$ | $34$ | $33$ | $( 1,33,10,25, 2,34,11,26, 3,18,12,27, 4,19,13,28, 5,20,14,29, 6,21,15,30, 7,22,16,31, 8,23,17,32, 9,24)$ |
| 34B9 | $34$ | $544$ | $34$ | $33$ | $( 1,27,15,24,12,21, 9,18, 6,32, 3,29,17,26,14,23,11,20, 8,34, 5,31, 2,28,16,25,13,22,10,19, 7,33, 4,30)$ |
| 34C1 | $34$ | $544$ | $34$ | $33$ | $( 1,31, 3,24, 5,34, 7,27, 9,20,11,30,13,23,15,33,17,26, 2,19, 4,29, 6,22, 8,32,10,25,12,18,14,28,16,21)$ |
| 34C3 | $34$ | $544$ | $34$ | $33$ | $( 1,24, 7,20,13,33, 2,29, 8,25,14,21, 3,34, 9,30,15,26, 4,22,10,18,16,31, 5,27,11,23,17,19, 6,32,12,28)$ |
| 34C7 | $34$ | $544$ | $34$ | $33$ | $( 1,27,15,29,12,31, 9,33, 6,18, 3,20,17,22,14,24,11,26, 8,28, 5,30, 2,32,16,34,13,19,10,21, 7,23, 4,25)$ |
| 34C9 | $34$ | $544$ | $34$ | $33$ | $( 1,20, 2,25, 3,30, 4,18, 5,23, 6,28, 7,33, 8,21, 9,26,10,31,11,19,12,24,13,29,14,34,15,22,16,27,17,32)$ |
Malle's constant $a(G)$: $1/8$
magma: ConjugacyClasses(G);
Character table
40 x 40 character tablemagma: CharacterTable(G);
Regular extensions
Data not computed