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Magma
magma: G := TransitiveGroup(34, 39);
Group invariants
Abstract group: | $D_{17}^2.Q_8$ | 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 $t$: | $39$ | 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: | $(1,25,3,28,12,33,10,30)(2,18,16,22,11,23,14,19)(4,21,8,27,9,20,5,31)(6,24,17,32,7,34,13,26)(15,29)$, $(1,33,10,22,12,29,3,23)(2,28,14,19,11,34,16,26)(4,18,5,30,9,27,8,32)(6,25,13,24,7,20,17,21)(15,31)$ | 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, $C_2^2$ $8$: $D_{4}$, $C_4\times C_2$, $Q_8$ $16$: $C_4:C_4$ $32$: 16T49 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 | $2^{8},1^{18}$ | $34$ | $2$ | $8$ | $( 1, 4)( 2, 3)( 5,17)( 6,16)( 7,15)( 8,14)( 9,13)(10,12)$ |
2B | $2^{16},1^{2}$ | $289$ | $2$ | $16$ | $( 1,10)( 2, 9)( 3, 8)( 4, 7)( 5, 6)(11,17)(12,16)(13,15)(18,20)(21,34)(22,33)(23,32)(24,31)(25,30)(26,29)(27,28)$ |
4A1 | $4^{8},1^{2}$ | $289$ | $4$ | $24$ | $( 1,13,10,15)( 2,17, 9,11)( 3, 4, 8, 7)( 5,12, 6,16)(18,32,20,23)(21,27,34,28)(22,31,33,24)(25,26,30,29)$ |
4A-1 | $4^{8},1^{2}$ | $289$ | $4$ | $24$ | $( 1,15,10,13)( 2,11, 9,17)( 3, 7, 8, 4)( 5,16, 6,12)(18,23,20,32)(21,28,34,27)(22,24,33,31)(25,29,30,26)$ |
4B | $4^{8},1^{2}$ | $578$ | $4$ | $24$ | $( 1, 6, 9, 4)( 2,10, 8,17)( 3,14, 7,13)(11,12,16,15)(18,32,27,30)(19,28,26,34)(20,24,25,21)(22,33,23,29)$ |
8A1 | $8^{4},1^{2}$ | $578$ | $8$ | $28$ | $( 1,14, 6, 7, 9,13, 4, 3)( 2,16,10,15, 8,11,17,12)(18,29,32,22,27,33,30,23)(19,20,28,24,26,25,34,21)$ |
8A-1 | $8^{4},1^{2}$ | $578$ | $8$ | $28$ | $( 1, 3, 4,13, 9, 7, 6,14)( 2,12,17,11, 8,15,10,16)(18,23,30,33,27,22,32,29)(19,21,34,25,26,24,28,20)$ |
8B1 | $8^{4},1^{2}$ | $578$ | $8$ | $28$ | $( 1, 4,15,10, 3,17, 6,11)( 5,13,14,12,16, 8, 7, 9)(18,30,24,27,34,22,28,25)(19,21,20,29,33,31,32,23)$ |
8B3 | $8^{4},1^{2}$ | $578$ | $8$ | $28$ | $( 1,10, 6, 4, 3,11,15,17)( 5,12, 7,13,16, 9,14, 8)(18,27,28,30,34,25,24,22)(19,29,32,21,33,23,20,31)$ |
8C1 | $8^{4},2$ | $1156$ | $8$ | $29$ | $( 1,25,13,26,10,30,15,29)( 2,18,17,32, 9,20,11,23)( 3,28, 4,21, 8,27, 7,34)( 5,31,12,33, 6,24,16,22)(14,19)$ |
8C-1 | $8^{4},2$ | $1156$ | $8$ | $29$ | $( 1,29,15,30,10,26,13,25)( 2,23,11,20, 9,32,17,18)( 3,34, 7,27, 8,21, 4,28)( 5,22,16,24, 6,33,12,31)(14,19)$ |
8D1 | $8^{4},2$ | $1156$ | $8$ | $29$ | $( 1,31, 3,21,11,32, 9,25)( 2,26, 7,18,10,20, 5,28)( 4,33,15,29, 8,30,14,34)( 6,23)(12,27,13,22,17,19,16,24)$ |
8D-1 | $8^{4},2$ | $1156$ | $8$ | $29$ | $( 1,25, 9,32,11,21, 3,31)( 2,28, 5,20,10,18, 7,26)( 4,34,14,30, 8,29,15,33)( 6,23)(12,24,16,19,17,22,13,27)$ |
17A1 | $17,1^{17}$ | $16$ | $17$ | $16$ | $(18,31,27,23,19,32,28,24,20,33,29,25,21,34,30,26,22)$ |
17A3 | $17,1^{17}$ | $16$ | $17$ | $16$ | $(18,23,28,33,21,26,31,19,24,29,34,22,27,32,20,25,30)$ |
17B1 | $17^{2}$ | $32$ | $17$ | $32$ | $( 1,11, 4,14, 7,17,10, 3,13, 6,16, 9, 2,12, 5,15, 8)(18,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19)$ |
17B3 | $17^{2}$ | $32$ | $17$ | $32$ | $( 1,14,10, 6, 2,15,11, 7, 3,16,12, 8, 4,17,13, 9, 5)(18,32,29,26,23,20,34,31,28,25,22,19,33,30,27,24,21)$ |
17C1 | $17^{2}$ | $32$ | $17$ | $32$ | $( 1,13, 8, 3,15,10, 5,17,12, 7, 2,14, 9, 4,16,11, 6)(18,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19)$ |
17C3 | $17^{2}$ | $32$ | $17$ | $32$ | $( 1,10, 2,11, 3,12, 4,13, 5,14, 6,15, 7,16, 8,17, 9)(18,32,29,26,23,20,34,31,28,25,22,19,33,30,27,24,21)$ |
17D1 | $17^{2}$ | $32$ | $17$ | $32$ | $( 1, 9,17, 8,16, 7,15, 6,14, 5,13, 4,12, 3,11, 2,10)(18,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19)$ |
17D2 | $17^{2}$ | $32$ | $17$ | $32$ | $( 1,17,16,15,14,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(18,33,31,29,27,25,23,21,19,34,32,30,28,26,24,22,20)$ |
17D3 | $17^{2}$ | $32$ | $17$ | $32$ | $( 1, 8,15, 5,12, 2, 9,16, 6,13, 3,10,17, 7,14, 4,11)(18,32,29,26,23,20,34,31,28,25,22,19,33,30,27,24,21)$ |
17D6 | $17^{2}$ | $32$ | $17$ | $32$ | $( 1,15,12, 9, 6, 3,17,14,11, 8, 5, 2,16,13,10, 7, 4)(18,29,23,34,28,22,33,27,21,32,26,20,31,25,19,30,24)$ |
34A1 | $17,2^{8},1$ | $272$ | $34$ | $24$ | $( 1, 4)( 2, 3)( 5,17)( 6,16)( 7,15)( 8,14)( 9,13)(10,12)(18,33,31,29,27,25,23,21,19,34,32,30,28,26,24,22,20)$ |
34A3 | $17,2^{8},1$ | $272$ | $34$ | $24$ | $( 1, 4)( 2, 3)( 5,17)( 6,16)( 7,15)( 8,14)( 9,13)(10,12)(18,29,23,34,28,22,33,27,21,32,26,20,31,25,19,30,24)$ |
Malle's constant $a(G)$: $1/8$
magma: ConjugacyClasses(G);
Character table
1A | 2A | 2B | 4A1 | 4A-1 | 4B | 8A1 | 8A-1 | 8B1 | 8B3 | 8C1 | 8C-1 | 8D1 | 8D-1 | 17A1 | 17A3 | 17B1 | 17B3 | 17C1 | 17C3 | 17D1 | 17D2 | 17D3 | 17D6 | 34A1 | 34A3 | ||
Size | 1 | 34 | 289 | 289 | 289 | 578 | 578 | 578 | 578 | 578 | 1156 | 1156 | 1156 | 1156 | 16 | 16 | 32 | 32 | 32 | 32 | 32 | 32 | 32 | 32 | 272 | 272 | |
2 P | 1A | 1A | 1A | 2B | 2B | 2B | 4B | 4B | 4B | 4B | 4A1 | 4A-1 | 4A1 | 4A-1 | 17A1 | 17A3 | 17B1 | 17B3 | 17C1 | 17C3 | 17D2 | 17D1 | 17D6 | 17D3 | 17A1 | 17A3 | |
17 P | 1A | 2A | 2B | 4A1 | 4A-1 | 4B | 8A1 | 8A-1 | 8B1 | 8B3 | 8C1 | 8C-1 | 8D1 | 8D-1 | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 2A | 2A | |
Type | |||||||||||||||||||||||||||
9248.y.1a | R | ||||||||||||||||||||||||||
9248.y.1b | R | ||||||||||||||||||||||||||
9248.y.1c | R | ||||||||||||||||||||||||||
9248.y.1d | R | ||||||||||||||||||||||||||
9248.y.1e1 | C | ||||||||||||||||||||||||||
9248.y.1e2 | C | ||||||||||||||||||||||||||
9248.y.1f1 | C | ||||||||||||||||||||||||||
9248.y.1f2 | C | ||||||||||||||||||||||||||
9248.y.2a | R | ||||||||||||||||||||||||||
9248.y.2b | S | ||||||||||||||||||||||||||
9248.y.2c1 | C | ||||||||||||||||||||||||||
9248.y.2c2 | C | ||||||||||||||||||||||||||
9248.y.2c3 | C | ||||||||||||||||||||||||||
9248.y.2c4 | C | ||||||||||||||||||||||||||
9248.y.16a1 | R | ||||||||||||||||||||||||||
9248.y.16a2 | R | ||||||||||||||||||||||||||
9248.y.16b1 | R | ||||||||||||||||||||||||||
9248.y.16b2 | R | ||||||||||||||||||||||||||
9248.y.32a1 | R | ||||||||||||||||||||||||||
9248.y.32a2 | R | ||||||||||||||||||||||||||
9248.y.32b1 | R | ||||||||||||||||||||||||||
9248.y.32b2 | R | ||||||||||||||||||||||||||
9248.y.32c1 | R | ||||||||||||||||||||||||||
9248.y.32c2 | R | ||||||||||||||||||||||||||
9248.y.32c3 | R | ||||||||||||||||||||||||||
9248.y.32c4 | R |
magma: CharacterTable(G);
Regular extensions
Data not computed