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Magma
magma: G := TransitiveGroup(34, 40);
Group invariants
| Abstract group: | $D_{17}^2.D_4$ | magma: IdentifyGroup(G);
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| Order: | $9248=2^{5} \cdot 17^{2}$ | 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 $t$: | $40$ | 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: | $(1,13,8,3,15,10,5,17,12,7,2,14,9,4,16,11,6)(18,33,25,27)(19,20,24,23)(21,28,22,32)(26,31,34,29)$, $(1,18,10,21,2,24,11,27,3,30,12,33,4,19,13,22,5,25,14,28,6,31,15,34,7,20,16,23,8,26,17,29,9,32)$ | 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, $C_2^2$ $8$: $D_{4}$ x 2, $C_4\times C_2$ $16$: $C_2^2:C_4$ $32$: $C_4\wr C_2$ 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 | $2^{8},1^{18}$ | $34$ | $2$ | $8$ | $(19,34)(20,33)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)$ |
| 2B | $2^{17}$ | $68$ | $2$ | $17$ | $( 1,25)( 2,31)( 3,20)( 4,26)( 5,32)( 6,21)( 7,27)( 8,33)( 9,22)(10,28)(11,34)(12,23)(13,29)(14,18)(15,24)(16,30)(17,19)$ |
| 2C | $2^{16},1^{2}$ | $289$ | $2$ | $16$ | $( 1,10)( 2, 9)( 3, 8)( 4, 7)( 5, 6)(11,17)(12,16)(13,15)(19,34)(20,33)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)$ |
| 4A1 | $4^{4},1^{18}$ | $34$ | $4$ | $12$ | $(19,31,34,22)(20,27,33,26)(21,23,32,30)(24,28,29,25)$ |
| 4A-1 | $4^{4},1^{18}$ | $34$ | $4$ | $12$ | $(19,22,34,31)(20,26,33,27)(21,30,32,23)(24,25,29,28)$ |
| 4B1 | $4^{8},1^{2}$ | $289$ | $4$ | $24$ | $( 1,13,10,15)( 2,17, 9,11)( 3, 4, 8, 7)( 5,12, 6,16)(19,22,34,31)(20,26,33,27)(21,30,32,23)(24,25,29,28)$ |
| 4B-1 | $4^{8},1^{2}$ | $289$ | $4$ | $24$ | $( 1,15,10,13)( 2,11, 9,17)( 3, 7, 8, 4)( 5,16, 6,12)(19,31,34,22)(20,27,33,26)(21,23,32,30)(24,28,29,25)$ |
| 4C | $4^{8},1^{2}$ | $578$ | $4$ | $24$ | $( 1,15,10,13)( 2,11, 9,17)( 3, 7, 8, 4)( 5,16, 6,12)(19,22,34,31)(20,26,33,27)(21,30,32,23)(24,25,29,28)$ |
| 4D1 | $4^{4},2^{8},1^{2}$ | $578$ | $4$ | $20$ | $( 1,10)( 2, 9)( 3, 8)( 4, 7)( 5, 6)(11,17)(12,16)(13,15)(19,31,34,22)(20,27,33,26)(21,23,32,30)(24,28,29,25)$ |
| 4D-1 | $4^{4},2^{8},1^{2}$ | $578$ | $4$ | $20$ | $( 1,10)( 2, 9)( 3, 8)( 4, 7)( 5, 6)(11,17)(12,16)(13,15)(19,22,34,31)(20,26,33,27)(21,30,32,23)(24,25,29,28)$ |
| 4E | $4^{8},2$ | $1156$ | $4$ | $25$ | $( 1,25,10,28)( 2,31, 9,22)( 3,20, 8,33)( 4,26, 7,27)( 5,32, 6,21)(11,34,17,19)(12,23,16,30)(13,29,15,24)(14,18)$ |
| 8A1 | $8^{4},2$ | $1156$ | $8$ | $29$ | $( 1,24,13,25,10,29,15,28)( 2,34,17,31, 9,19,11,22)( 3,27, 4,20, 8,26, 7,33)( 5,30,12,32, 6,23,16,21)(14,18)$ |
| 8A-1 | $8^{4},2$ | $1156$ | $8$ | $29$ | $( 1,24,15,28,10,29,13,25)( 2,34,11,22, 9,19,17,31)( 3,27, 7,33, 8,26, 4,20)( 5,30,16,21, 6,23,12,32)(14,18)$ |
| 17A1 | $17,1^{17}$ | $8$ | $17$ | $16$ | $(18,23,28,33,21,26,31,19,24,29,34,22,27,32,20,25,30)$ |
| 17A2 | $17,1^{17}$ | $8$ | $17$ | $16$ | $(18,28,21,31,24,34,27,20,30,23,33,26,19,29,22,32,25)$ |
| 17A3 | $17,1^{17}$ | $8$ | $17$ | $16$ | $(18,33,31,29,27,25,23,21,19,34,32,30,28,26,24,22,20)$ |
| 17A6 | $17,1^{17}$ | $8$ | $17$ | $16$ | $(18,31,27,23,19,32,28,24,20,33,29,25,21,34,30,26,22)$ |
| 17B1 | $17^{2}$ | $16$ | $17$ | $32$ | $( 1, 5, 9,13,17, 4, 8,12,16, 3, 7,11,15, 2, 6,10,14)(18,28,21,31,24,34,27,20,30,23,33,26,19,29,22,32,25)$ |
| 17B2 | $17^{2}$ | $16$ | $17$ | $32$ | $( 1, 9,17, 8,16, 7,15, 6,14, 5,13, 4,12, 3,11, 2,10)(18,21,24,27,30,33,19,22,25,28,31,34,20,23,26,29,32)$ |
| 17B3 | $17^{2}$ | $16$ | $17$ | $32$ | $( 1,13, 8, 3,15,10, 5,17,12, 7, 2,14, 9, 4,16,11, 6)(18,31,27,23,19,32,28,24,20,33,29,25,21,34,30,26,22)$ |
| 17B6 | $17^{2}$ | $16$ | $17$ | $32$ | $( 1, 8,15, 5,12, 2, 9,16, 6,13, 3,10,17, 7,14, 4,11)(18,27,19,28,20,29,21,30,22,31,23,32,24,33,25,34,26)$ |
| 17C1 | $17^{2}$ | $32$ | $17$ | $32$ | $( 1, 3, 5, 7, 9,11,13,15,17, 2, 4, 6, 8,10,12,14,16)(18,28,21,31,24,34,27,20,30,23,33,26,19,29,22,32,25)$ |
| 17C3 | $17^{2}$ | $32$ | $17$ | $32$ | $( 1, 9,17, 8,16, 7,15, 6,14, 5,13, 4,12, 3,11, 2,10)(18,31,27,23,19,32,28,24,20,33,29,25,21,34,30,26,22)$ |
| 17D1 | $17^{2}$ | $32$ | $17$ | $32$ | $( 1, 5, 9,13,17, 4, 8,12,16, 3, 7,11,15, 2, 6,10,14)(18,33,31,29,27,25,23,21,19,34,32,30,28,26,24,22,20)$ |
| 17D2 | $17^{2}$ | $32$ | $17$ | $32$ | $( 1,15,12, 9, 6, 3,17,14,11, 8, 5, 2,16,13,10, 7, 4)(18,24,30,19,25,31,20,26,32,21,27,33,22,28,34,23,29)$ |
| 17D3 | $17^{2}$ | $32$ | $17$ | $32$ | $( 1,11, 4,14, 7,17,10, 3,13, 6,16, 9, 2,12, 5,15, 8)(18,31,27,23,19,32,28,24,20,33,29,25,21,34,30,26,22)$ |
| 17D6 | $17^{2}$ | $32$ | $17$ | $32$ | $( 1, 9,17, 8,16, 7,15, 6,14, 5,13, 4,12, 3,11, 2,10)(18,26,34,25,33,24,32,23,31,22,30,21,29,20,28,19,27)$ |
| 34A1 | $17,2^{8},1$ | $136$ | $34$ | $24$ | $( 1, 8,15, 5,12, 2, 9,16, 6,13, 3,10,17, 7,14, 4,11)(18,26)(19,25)(20,24)(21,23)(27,34)(28,33)(29,32)(30,31)$ |
| 34A3 | $17,2^{8},1$ | $136$ | $34$ | $24$ | $( 1, 5, 9,13,17, 4, 8,12,16, 3, 7,11,15, 2, 6,10,14)(18,25)(19,24)(20,23)(21,22)(26,34)(27,33)(28,32)(29,31)$ |
| 34A7 | $17,2^{8},1$ | $136$ | $34$ | $24$ | $( 1,13, 8, 3,15,10, 5,17,12, 7, 2,14, 9, 4,16,11, 6)(18,22)(19,21)(23,34)(24,33)(25,32)(26,31)(27,30)(28,29)$ |
| 34A9 | $17,2^{8},1$ | $136$ | $34$ | $24$ | $( 1, 9,17, 8,16, 7,15, 6,14, 5,13, 4,12, 3,11, 2,10)(18,32)(19,31)(20,30)(21,29)(22,28)(23,27)(24,26)(33,34)$ |
| 34B1 | $34$ | $272$ | $34$ | $33$ | $( 1,25,14,18,10,28, 6,21, 2,31,15,24,11,34, 7,27, 3,20,16,30,12,23, 8,33, 4,26,17,19,13,29, 9,22, 5,32)$ |
| 34B3 | $34$ | $272$ | $34$ | $33$ | $( 1,25, 6,21,11,34,16,30, 4,26, 9,22,14,18, 2,31, 7,27,12,23,17,19, 5,32,10,28,15,24, 3,20, 8,33,13,29)$ |
| 34B7 | $34$ | $272$ | $34$ | $33$ | $( 1,25,12,23, 6,21,17,19,11,34, 5,32,16,30,10,28, 4,26,15,24, 9,22, 3,20,14,18, 8,33, 2,31,13,29, 7,27)$ |
| 34B9 | $34$ | $272$ | $34$ | $33$ | $( 1,25,16,30,14,18,12,23,10,28, 8,33, 6,21, 4,26, 2,31,17,19,15,24,13,29,11,34, 9,22, 7,27, 5,32, 3,20)$ |
| 68A1 | $17,4^{4},1$ | $136$ | $68$ | $28$ | $( 1, 5, 9,13,17, 4, 8,12,16, 3, 7,11,15, 2, 6,10,14)(18,24,31,25)(19,28,30,21)(20,32,29,34)(22,23,27,26)$ |
| 68A-1 | $17,4^{4},1$ | $136$ | $68$ | $28$ | $( 1, 9,17, 8,16, 7,15, 6,14, 5,13, 4,12, 3,11, 2,10)(18,30,27,32)(19,34,26,28)(20,21,25,24)(22,29,23,33)$ |
| 68A3 | $17,4^{4},1$ | $136$ | $68$ | $28$ | $( 1, 8,15, 5,12, 2, 9,16, 6,13, 3,10,17, 7,14, 4,11)(18,20,28,26)(19,24,27,22)(21,32,25,31)(29,30,34,33)$ |
| 68A-3 | $17,4^{4},1$ | $136$ | $68$ | $28$ | $( 1,13, 8, 3,15,10, 5,17,12, 7, 2,14, 9, 4,16,11, 6)(18,34,21,22)(19,30,20,26)(23,31,33,25)(24,27,32,29)$ |
| 68A7 | $17,4^{4},1$ | $136$ | $68$ | $28$ | $( 1,13, 8, 3,15,10, 5,17,12, 7, 2,14, 9, 4,16,11, 6)(18,19,23,22)(20,27,21,31)(24,26,34,32)(25,30,33,28)$ |
| 68A-7 | $17,4^{4},1$ | $136$ | $68$ | $28$ | $( 1, 9,17, 8,16, 7,15, 6,14, 5,13, 4,12, 3,11, 2,10)(18,23,20,32)(21,28,34,27)(22,24,33,31)(25,29,30,26)$ |
| 68A9 | $17,4^{4},1$ | $136$ | $68$ | $28$ | $( 1, 8,15, 5,12, 2, 9,16, 6,13, 3,10,17, 7,14, 4,11)(18,33,24,26)(19,29,23,30)(20,25,22,34)(27,31,32,28)$ |
| 68A-9 | $17,4^{4},1$ | $136$ | $68$ | $28$ | $( 1, 5, 9,13,17, 4, 8,12,16, 3, 7,11,15, 2, 6,10,14)(18,29,19,25)(20,21,34,33)(22,30,32,24)(23,26,31,28)$ |
Malle's constant $a(G)$: $1/8$
magma: ConjugacyClasses(G);
Character table
44 x 44 character tablemagma: CharacterTable(G);
Regular extensions
Data not computed