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Magma
magma: G := TransitiveGroup(34, 20);
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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| Group: | $C_{17}^2:Q_{16}$ | ||
| Parity: | $-1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(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
|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 | Representative |
| $ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ | |
| $ 17, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $16$ | $17$ | $( 1, 4, 7,10,13,16, 2, 5, 8,11,14,17, 3, 6, 9,12,15)$ | |
| $ 17, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $16$ | $17$ | $( 1,10, 2,11, 3,12, 4,13, 5,14, 6,15, 7,16, 8,17, 9)$ | |
| $ 17, 17 $ | $16$ | $17$ | $( 1, 4, 7,10,13,16, 2, 5, 8,11,14,17, 3, 6, 9,12,15)(18,26,34,25,33,24,32,23, 31,22,30,21,29,20,28,19,27)$ | |
| $ 17, 17 $ | $16$ | $17$ | $( 1,10, 2,11, 3,12, 4,13, 5,14, 6,15, 7,16, 8,17, 9)(18,26,34,25,33,24,32,23, 31,22,30,21,29,20,28,19,27)$ | |
| $ 17, 17 $ | $16$ | $17$ | $( 1,11, 4,14, 7,17,10, 3,13, 6,16, 9, 2,12, 5,15, 8)(18,26,34,25,33,24,32,23, 31,22,30,21,29,20,28,19,27)$ | |
| $ 17, 17 $ | $16$ | $17$ | $( 1,14,10, 6, 2,15,11, 7, 3,16,12, 8, 4,17,13, 9, 5)(18,26,34,25,33,24,32,23, 31,22,30,21,29,20,28,19,27)$ | |
| $ 17, 17 $ | $16$ | $17$ | $( 1, 6,11,16, 4, 9,14, 2, 7,12,17, 5,10,15, 3, 8,13)(18,26,34,25,33,24,32,23, 31,22,30,21,29,20,28,19,27)$ | |
| $ 17, 17 $ | $16$ | $17$ | $( 1,16,14,12,10, 8, 6, 4, 2,17,15,13,11, 9, 7, 5, 3)(18,26,34,25,33,24,32,23, 31,22,30,21,29,20,28,19,27)$ | |
| $ 17, 17 $ | $16$ | $17$ | $( 1,12, 6,17,11, 5,16,10, 4,15, 9, 3,14, 8, 2,13, 7)(18,26,34,25,33,24,32,23, 31,22,30,21,29,20,28,19,27)$ | |
| $ 17, 17 $ | $16$ | $17$ | $( 1,17,16,15,14,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(18,26,34,25,33,24,32,23, 31,22,30,21,29,20,28,19,27)$ | |
| $ 17, 17 $ | $16$ | $17$ | $( 1, 9,17, 8,16, 7,15, 6,14, 5,13, 4,12, 3,11, 2,10)(18,26,34,25,33,24,32,23, 31,22,30,21,29,20,28,19,27)$ | |
| $ 17, 17 $ | $16$ | $17$ | $( 1, 5, 9,13,17, 4, 8,12,16, 3, 7,11,15, 2, 6,10,14)(18,26,34,25,33,24,32,23, 31,22,30,21,29,20,28,19,27)$ | |
| $ 17, 17 $ | $16$ | $17$ | $( 1, 3, 5, 7, 9,11,13,15,17, 2, 4, 6, 8,10,12,14,16)(18,26,34,25,33,24,32,23, 31,22,30,21,29,20,28,19,27)$ | |
| $ 17, 17 $ | $16$ | $17$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17)(18,26,34,25,33,24,32,23, 31,22,30,21,29,20,28,19,27)$ | |
| $ 17, 17 $ | $16$ | $17$ | $( 1,10, 2,11, 3,12, 4,13, 5,14, 6,15, 7,16, 8,17, 9)(18,25,32,22,29,19,26,33, 23,30,20,27,34,24,31,21,28)$ | |
| $ 17, 17 $ | $16$ | $17$ | $( 1,14,10, 6, 2,15,11, 7, 3,16,12, 8, 4,17,13, 9, 5)(18,25,32,22,29,19,26,33, 23,30,20,27,34,24,31,21,28)$ | |
| $ 17, 17 $ | $16$ | $17$ | $( 1,16,14,12,10, 8, 6, 4, 2,17,15,13,11, 9, 7, 5, 3)(18,25,32,22,29,19,26,33, 23,30,20,27,34,24,31,21,28)$ | |
| $ 17, 17 $ | $16$ | $17$ | $( 1,17,16,15,14,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(18,25,32,22,29,19,26,33, 23,30,20,27,34,24,31,21,28)$ | |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $289$ | $2$ | $( 2,17)( 3,16)( 4,15)( 5,14)( 6,13)( 7,12)( 8,11)( 9,10)(19,34)(20,33)(21,32) (22,31)(23,30)(24,29)(25,28)(26,27)$ | |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 1, 1 $ | $578$ | $4$ | $( 2, 5,17,14)( 3, 9,16,10)( 4,13,15, 6)( 7, 8,12,11)(19,31,34,22)(20,27,33,26) (21,23,32,30)(24,28,29,25)$ | |
| $ 8, 8, 8, 8, 1, 1 $ | $578$ | $8$ | $( 2, 3, 5, 9,17,16,14,10)( 4, 7,13, 8,15,12, 6,11)(19,27,31,33,34,26,22,20) (21,28,23,29,32,25,30,24)$ | |
| $ 8, 8, 8, 8, 1, 1 $ | $578$ | $8$ | $( 2,16, 5,10,17, 3,14, 9)( 4,12,13,11,15, 7, 6, 8)(19,26,31,20,34,27,22,33) (21,25,23,24,32,28,30,29)$ | |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 2 $ | $1156$ | $4$ | $( 1,21, 9,31)( 2,18, 8,34)( 3,32, 7,20)( 4,29, 6,23)( 5,26)(10,28,17,24) (11,25,16,27)(12,22,15,30)(13,19,14,33)$ | |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 2 $ | $1156$ | $4$ | $( 1,28,16,31)( 2,18,15,24)( 3,25,14,34)( 4,32,13,27)( 5,22,12,20)( 6,29,11,30) ( 7,19,10,23)( 8,26, 9,33)(17,21)$ |
magma: ConjugacyClasses(G);
Group invariants
| 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 | ||
| Label: | 4624.u | magma: IdentifyGroup(G);
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| Character table: |
| Size | |
| 2 P | |
| 17 P | |
| Type |
magma: CharacterTable(G);