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Magma
magma: G := TransitiveGroup(34, 20);
Group invariants
Abstract group: | $C_{17}^2:Q_{16}$ | magma: IdentifyGroup(G);
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Order: | $4624=2^{4} \cdot 17^{2}$ | 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 $t$: | $20$ | 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: | $(1,3,7,15,14,12,8,17)(2,5,11,6,13,10,4,9)(18,31,29,28,19,23,25,26)(20,32,21,24,34,22,33,30)$, $(1,21,9,31)(2,18,8,34)(3,32,7,20)(4,29,6,23)(5,26)(10,28,17,24)(11,25,16,27)(12,22,15,30)(13,19,14,33)$ | magma: Generators(G);
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Low degree resolvents
$\card{(G/N)}$ Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_2^2$ $8$: $D_{4}$ $16$: $Q_{16}$ 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 | $2^{16},1^{2}$ | $289$ | $2$ | $16$ | $( 1,12)( 2,11)( 3,10)( 4, 9)( 5, 8)( 6, 7)(13,17)(14,16)(18,28)(19,27)(20,26)(21,25)(22,24)(29,34)(30,33)(31,32)$ |
4A | $4^{8},1^{2}$ | $578$ | $4$ | $24$ | $( 1, 8, 2,12)( 3,16,17, 4)( 5, 7,15,13)( 6,11,14, 9)(18,28,22,29)(19,24,21,33)(23,25,34,32)(26,30,31,27)$ |
4B | $4^{8},2$ | $1156$ | $4$ | $25$ | $( 1,19,12,27)( 2,29,11,34)( 3,22,10,24)( 4,32, 9,31)( 5,25, 8,21)( 6,18, 7,28)(13,20,17,26)(14,30,16,33)(15,23)$ |
4C | $4^{8},2$ | $1156$ | $4$ | $25$ | $( 1,30, 8,26)( 2,27, 7,29)( 3,24, 6,32)( 4,21, 5,18)( 9,23,17,33)(10,20,16,19)(11,34,15,22)(12,31,14,25)(13,28)$ |
8A1 | $8^{4},1^{2}$ | $578$ | $8$ | $28$ | $( 1,11, 8,14, 2, 9,12, 6)( 3, 7,16,15,17,13, 4, 5)(18,21,28,33,22,19,29,24)(23,27,25,26,34,30,32,31)$ |
8A3 | $8^{4},1^{2}$ | $578$ | $8$ | $28$ | $( 1,14,12,11, 2, 6, 8, 9)( 3,15, 4, 7,17, 5,16,13)(18,33,29,21,22,24,28,19)(23,26,32,27,34,31,25,30)$ |
17A1 | $17,1^{17}$ | $16$ | $17$ | $16$ | $( 1, 9,17, 8,16, 7,15, 6,14, 5,13, 4,12, 3,11, 2,10)$ |
17A3 | $17,1^{17}$ | $16$ | $17$ | $16$ | $( 1, 8,15, 5,12, 2, 9,16, 6,13, 3,10,17, 7,14, 4,11)$ |
17B1 | $17^{2}$ | $16$ | $17$ | $32$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17)(18,32,29,26,23,20,34,31,28,25,22,19,33,30,27,24,21)$ |
17B2 | $17^{2}$ | $16$ | $17$ | $32$ | $( 1, 3, 5, 7, 9,11,13,15,17, 2, 4, 6, 8,10,12,14,16)(18,29,23,34,28,22,33,27,21,32,26,20,31,25,19,30,24)$ |
17B3 | $17^{2}$ | $16$ | $17$ | $32$ | $( 1, 4, 7,10,13,16, 2, 5, 8,11,14,17, 3, 6, 9,12,15)(18,26,34,25,33,24,32,23,31,22,30,21,29,20,28,19,27)$ |
17B6 | $17^{2}$ | $16$ | $17$ | $32$ | $( 1, 7,13, 2, 8,14, 3, 9,15, 4,10,16, 5,11,17, 6,12)(18,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19)$ |
17C1 | $17^{2}$ | $16$ | $17$ | $32$ | $( 1,10, 2,11, 3,12, 4,13, 5,14, 6,15, 7,16, 8,17, 9)(18,32,29,26,23,20,34,31,28,25,22,19,33,30,27,24,21)$ |
17C2 | $17^{2}$ | $16$ | $17$ | $32$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17)(18,29,23,34,28,22,33,27,21,32,26,20,31,25,19,30,24)$ |
17C3 | $17^{2}$ | $16$ | $17$ | $32$ | $( 1,11, 4,14, 7,17,10, 3,13, 6,16, 9, 2,12, 5,15, 8)(18,26,34,25,33,24,32,23,31,22,30,21,29,20,28,19,27)$ |
17C6 | $17^{2}$ | $16$ | $17$ | $32$ | $( 1,12, 6,17,11, 5,16,10, 4,15, 9, 3,14, 8, 2,13, 7)(18,26,34,25,33,24,32,23,31,22,30,21,29,20,28,19,27)$ |
17D1 | $17^{2}$ | $16$ | $17$ | $32$ | $( 1,11, 4,14, 7,17,10, 3,13, 6,16, 9, 2,12, 5,15, 8)(18,29,23,34,28,22,33,27,21,32,26,20,31,25,19,30,24)$ |
17D2 | $17^{2}$ | $16$ | $17$ | $32$ | $( 1, 4, 7,10,13,16, 2, 5, 8,11,14,17, 3, 6, 9,12,15)(18,23,28,33,21,26,31,19,24,29,34,22,27,32,20,25,30)$ |
17D3 | $17^{2}$ | $16$ | $17$ | $32$ | $( 1, 8,15, 5,12, 2, 9,16, 6,13, 3,10,17, 7,14, 4,11)(18,32,29,26,23,20,34,31,28,25,22,19,33,30,27,24,21)$ |
17D4 | $17^{2}$ | $16$ | $17$ | $32$ | $( 1,13, 8, 3,15,10, 5,17,12, 7, 2,14, 9, 4,16,11, 6)(18,23,28,33,21,26,31,19,24,29,34,22,27,32,20,25,30)$ |
17D5 | $17^{2}$ | $16$ | $17$ | $32$ | $( 1,12, 6,17,11, 5,16,10, 4,15, 9, 3,14, 8, 2,13, 7)(18,23,28,33,21,26,31,19,24,29,34,22,27,32,20,25,30)$ |
17D6 | $17^{2}$ | $16$ | $17$ | $32$ | $( 1, 3, 5, 7, 9,11,13,15,17, 2, 4, 6, 8,10,12,14,16)(18,26,34,25,33,24,32,23,31,22,30,21,29,20,28,19,27)$ |
17D7 | $17^{2}$ | $16$ | $17$ | $32$ | $( 1, 7,13, 2, 8,14, 3, 9,15, 4,10,16, 5,11,17, 6,12)(18,32,29,26,23,20,34,31,28,25,22,19,33,30,27,24,21)$ |
17D8 | $17^{2}$ | $16$ | $17$ | $32$ | $( 1, 8,15, 5,12, 2, 9,16, 6,13, 3,10,17, 7,14, 4,11)(18,28,21,31,24,34,27,20,30,23,33,26,19,29,22,32,25)$ |
Malle's constant $a(G)$: $1/16$
magma: ConjugacyClasses(G);
Character table
1A | 2A | 4A | 4B | 4C | 8A1 | 8A3 | 17A1 | 17A3 | 17B1 | 17B2 | 17B3 | 17B6 | 17C1 | 17C2 | 17C3 | 17C6 | 17D1 | 17D2 | 17D3 | 17D4 | 17D5 | 17D6 | 17D7 | 17D8 | ||
Size | 1 | 289 | 578 | 1156 | 1156 | 578 | 578 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | |
2 P | 1A | 1A | 2A | 2A | 2A | 4A | 4A | 17A1 | 17A3 | 17B2 | 17B1 | 17B6 | 17B3 | 17C2 | 17C1 | 17C6 | 17C3 | 17D2 | 17D4 | 17D6 | 17D8 | 17D7 | 17D5 | 17D3 | 17D1 | |
17 P | 1A | 2A | 4A | 4B | 4C | 8A1 | 8A3 | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 1A | |
Type | ||||||||||||||||||||||||||
4624.u.1a | R | |||||||||||||||||||||||||
4624.u.1b | R | |||||||||||||||||||||||||
4624.u.1c | R | |||||||||||||||||||||||||
4624.u.1d | R | |||||||||||||||||||||||||
4624.u.2a | R | |||||||||||||||||||||||||
4624.u.2b1 | S | |||||||||||||||||||||||||
4624.u.2b2 | S | |||||||||||||||||||||||||
4624.u.16a1 | R | |||||||||||||||||||||||||
4624.u.16a2 | R | |||||||||||||||||||||||||
4624.u.16b1 | R | |||||||||||||||||||||||||
4624.u.16b2 | R | |||||||||||||||||||||||||
4624.u.16b3 | R | |||||||||||||||||||||||||
4624.u.16b4 | R | |||||||||||||||||||||||||
4624.u.16c1 | R | |||||||||||||||||||||||||
4624.u.16c2 | R | |||||||||||||||||||||||||
4624.u.16c3 | R | |||||||||||||||||||||||||
4624.u.16c4 | R | |||||||||||||||||||||||||
4624.u.16d1 | R | |||||||||||||||||||||||||
4624.u.16d2 | R | |||||||||||||||||||||||||
4624.u.16d3 | R | |||||||||||||||||||||||||
4624.u.16d4 | R | |||||||||||||||||||||||||
4624.u.16d5 | R | |||||||||||||||||||||||||
4624.u.16d6 | R | |||||||||||||||||||||||||
4624.u.16d7 | R | |||||||||||||||||||||||||
4624.u.16d8 | R |
magma: CharacterTable(G);
Regular extensions
Data not computed