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Magma
magma: G := TransitiveGroup(34, 33);
Group invariants
Abstract group: | $C_{17}^2:Q_{32}$ | 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 $t$: | $33$ | 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: | $(1,20,16,22)(2,19,15,23)(3,18,14,24)(4,34,13,25)(5,33,12,26)(6,32,11,27)(7,31,10,28)(8,30,9,29)(17,21)$, $(1,18,9,24)(2,23,8,19)(3,28,7,31)(4,33,6,26)(5,21)(10,29,17,30)(11,34,16,25)(12,22,15,20)(13,27,14,32)$ | magma: Generators(G);
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Low degree resolvents
$\card{(G/N)}$ Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_2^2$ $8$: $D_{4}$ $16$: $D_{8}$ $32$: 32T51 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 | $2^{16},1^{2}$ | $289$ | $2$ | $16$ | $( 1,11)( 2,10)( 3, 9)( 4, 8)( 5, 7)(12,17)(13,16)(14,15)(18,25)(19,24)(20,23)(21,22)(26,34)(27,33)(28,32)(29,31)$ |
4A | $4^{8},1^{2}$ | $578$ | $4$ | $24$ | $( 1, 9,11, 3)( 2, 5,10, 7)( 4,14, 8,15)(12,16,17,13)(18,33,25,27)(19,20,24,23)(21,28,22,32)(26,31,34,29)$ |
4B | $4^{8},2$ | $2312$ | $4$ | $25$ | $( 1,21)( 2,33,17,26)( 3,28,16,31)( 4,23,15,19)( 5,18,14,24)( 6,30,13,29)( 7,25,12,34)( 8,20,11,22)( 9,32,10,27)$ |
4C | $4^{8},2$ | $2312$ | $4$ | $25$ | $( 1,18)( 2,34,17,19)( 3,33,16,20)( 4,32,15,21)( 5,31,14,22)( 6,30,13,23)( 7,29,12,24)( 8,28,11,25)( 9,27,10,26)$ |
8A1 | $8^{4},1^{2}$ | $578$ | $8$ | $28$ | $( 1,17, 9,13,11,12, 3,16)( 2, 8, 5,15,10, 4, 7,14)(18,20,33,24,25,23,27,19)(21,31,28,34,22,29,32,26)$ |
8A3 | $8^{4},1^{2}$ | $578$ | $8$ | $28$ | $( 1,12, 9,16,11,17, 3,13)( 2, 4, 5,14,10, 8, 7,15)(18,23,33,19,25,20,27,24)(21,29,28,26,22,31,32,34)$ |
16A1 | $16^{2},1^{2}$ | $578$ | $16$ | $30$ | $( 1,15,17,10, 9, 4,13, 7,11,14,12, 2, 3, 8,16, 5)(18,31,20,28,33,34,24,22,25,29,23,32,27,26,19,21)$ |
16A3 | $16^{2},1^{2}$ | $578$ | $16$ | $30$ | $( 1, 8,12, 7, 9,15,16, 2,11, 4,17, 5, 3,14,13,10)(18,26,23,22,33,31,19,32,25,34,20,21,27,29,24,28)$ |
16A5 | $16^{2},1^{2}$ | $578$ | $16$ | $30$ | $( 1, 4,12, 5, 9,14,16,10,11, 8,17, 7, 3,15,13, 2)(18,34,23,21,33,29,19,28,25,26,20,22,27,31,24,32)$ |
16A7 | $16^{2},1^{2}$ | $578$ | $16$ | $30$ | $( 1,14,17, 2, 9, 8,13, 5,11,15,12,10, 3, 4,16, 7)(18,29,20,32,33,26,24,21,25,31,23,28,27,34,19,22)$ |
17A | $17,1^{17}$ | $32$ | $17$ | $16$ | $( 1, 3, 5, 7, 9,11,13,15,17, 2, 4, 6, 8,10,12,14,16)$ |
17B1 | $17^{2}$ | $32$ | $17$ | $32$ | $( 1,10, 2,11, 3,12, 4,13, 5,14, 6,15, 7,16, 8,17, 9)(18,23,28,33,21,26,31,19,24,29,34,22,27,32,20,25,30)$ |
17B2 | $17^{2}$ | $32$ | $17$ | $32$ | $( 1,17,16,15,14,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(18,28,21,31,24,34,27,20,30,23,33,26,19,29,22,32,25)$ |
17B3 | $17^{2}$ | $32$ | $17$ | $32$ | $( 1,13, 8, 3,15,10, 5,17,12, 7, 2,14, 9, 4,16,11, 6)(18,31,27,23,19,32,28,24,20,33,29,25,21,34,30,26,22)$ |
17B6 | $17^{2}$ | $32$ | $17$ | $32$ | $( 1, 7,13, 2, 8,14, 3, 9,15, 4,10,16, 5,11,17, 6,12)(18,33,31,29,27,25,23,21,19,34,32,30,28,26,24,22,20)$ |
17C1 | $17^{2}$ | $32$ | $17$ | $32$ | $( 1, 8,15, 5,12, 2, 9,16, 6,13, 3,10,17, 7,14, 4,11)(18,23,28,33,21,26,31,19,24,29,34,22,27,32,20,25,30)$ |
17C2 | $17^{2}$ | $32$ | $17$ | $32$ | $( 1, 9,17, 8,16, 7,15, 6,14, 5,13, 4,12, 3,11, 2,10)(18,31,27,23,19,32,28,24,20,33,29,25,21,34,30,26,22)$ |
17C3 | $17^{2}$ | $32$ | $17$ | $32$ | $( 1,15,12, 9, 6, 3,17,14,11, 8, 5, 2,16,13,10, 7, 4)(18,28,21,31,24,34,27,20,30,23,33,26,19,29,22,32,25)$ |
17C6 | $17^{2}$ | $32$ | $17$ | $32$ | $( 1, 5, 9,13,17, 4, 8,12,16, 3, 7,11,15, 2, 6,10,14)(18,33,31,29,27,25,23,21,19,34,32,30,28,26,24,22,20)$ |
Malle's constant $a(G)$: $1/16$
magma: ConjugacyClasses(G);
Character table
1A | 2A | 4A | 4B | 4C | 8A1 | 8A3 | 16A1 | 16A3 | 16A5 | 16A7 | 17A | 17B1 | 17B2 | 17B3 | 17B6 | 17C1 | 17C2 | 17C3 | 17C6 | ||
Size | 1 | 289 | 578 | 2312 | 2312 | 578 | 578 | 578 | 578 | 578 | 578 | 32 | 32 | 32 | 32 | 32 | 32 | 32 | 32 | 32 | |
2 P | 1A | 1A | 2A | 2A | 2A | 4A | 4A | 8A1 | 8A3 | 8A3 | 8A1 | 17A | 17B2 | 17B1 | 17B6 | 17B3 | 17C2 | 17C3 | 17C1 | 17C6 | |
17 P | 1A | 2A | 4A | 4B | 4C | 8A1 | 8A3 | 16A1 | 16A3 | 16A5 | 16A7 | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 1A | |
Type |
magma: CharacterTable(G);
Regular extensions
Data not computed