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Magma
magma: G := TransitiveGroup(34, 1);
Group action invariants
| Degree $n$: | $34$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $1$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_{34}$ | ||
| Parity: | $-1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $34$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,25,16,5,30,20,9,33,23,14,4,27,18,7,31,21,12,2,26,15,6,29,19,10,34,24,13,3,28,17,8,32,22,11) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $17$: $C_{17}$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 17: $C_{17}$
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 $ | $1$ | $1$ | $()$ | |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)(29,30)(31,32)(33,34)$ | |
| $ 34 $ | $1$ | $34$ | $( 1, 3, 6, 7, 9,11,13,15,18,20,22,24,26,27,30,32,34, 2, 4, 5, 8,10,12,14,16, 17,19,21,23,25,28,29,31,33)$ | |
| $ 17, 17 $ | $1$ | $17$ | $( 1, 4, 6, 8, 9,12,13,16,18,19,22,23,26,28,30,31,34)( 2, 3, 5, 7,10,11,14,15, 17,20,21,24,25,27,29,32,33)$ | |
| $ 34 $ | $1$ | $34$ | $( 1, 5, 9,14,18,21,26,29,34, 3, 8,11,16,20,23,27,31, 2, 6,10,13,17,22,25,30, 33, 4, 7,12,15,19,24,28,32)$ | |
| $ 17, 17 $ | $1$ | $17$ | $( 1, 6, 9,13,18,22,26,30,34, 4, 8,12,16,19,23,28,31)( 2, 5,10,14,17,21,25,29, 33, 3, 7,11,15,20,24,27,32)$ | |
| $ 34 $ | $1$ | $34$ | $( 1, 7,13,20,26,32, 4,10,16,21,28,33, 6,11,18,24,30, 2, 8,14,19,25,31, 3, 9, 15,22,27,34, 5,12,17,23,29)$ | |
| $ 17, 17 $ | $1$ | $17$ | $( 1, 8,13,19,26,31, 4, 9,16,22,28,34, 6,12,18,23,30)( 2, 7,14,20,25,32, 3,10, 15,21,27,33, 5,11,17,24,29)$ | |
| $ 17, 17 $ | $1$ | $17$ | $( 1, 9,18,26,34, 8,16,23,31, 6,13,22,30, 4,12,19,28)( 2,10,17,25,33, 7,15,24, 32, 5,14,21,29, 3,11,20,27)$ | |
| $ 34 $ | $1$ | $34$ | $( 1,10,18,25,34, 7,16,24,31, 5,13,21,30, 3,12,20,28, 2, 9,17,26,33, 8,15,23, 32, 6,14,22,29, 4,11,19,27)$ | |
| $ 34 $ | $1$ | $34$ | $( 1,11,22,32, 8,17,28, 3,13,24,34,10,19,29, 6,15,26, 2,12,21,31, 7,18,27, 4, 14,23,33, 9,20,30, 5,16,25)$ | |
| $ 17, 17 $ | $1$ | $17$ | $( 1,12,22,31, 8,18,28, 4,13,23,34, 9,19,30, 6,16,26)( 2,11,21,32, 7,17,27, 3, 14,24,33,10,20,29, 5,15,25)$ | |
| $ 17, 17 $ | $1$ | $17$ | $( 1,13,26, 4,16,28, 6,18,30, 8,19,31, 9,22,34,12,23)( 2,14,25, 3,15,27, 5,17, 29, 7,20,32,10,21,33,11,24)$ | |
| $ 34 $ | $1$ | $34$ | $( 1,14,26, 3,16,27, 6,17,30, 7,19,32, 9,21,34,11,23, 2,13,25, 4,15,28, 5,18, 29, 8,20,31,10,22,33,12,24)$ | |
| $ 34 $ | $1$ | $34$ | $( 1,15,30,10,23, 3,18,32,12,25, 6,20,34,14,28, 7,22, 2,16,29, 9,24, 4,17,31, 11,26, 5,19,33,13,27, 8,21)$ | |
| $ 17, 17 $ | $1$ | $17$ | $( 1,16,30, 9,23, 4,18,31,12,26, 6,19,34,13,28, 8,22)( 2,15,29,10,24, 3,17,32, 11,25, 5,20,33,14,27, 7,21)$ | |
| $ 34 $ | $1$ | $34$ | $( 1,17,34,15,31,14,30,11,28,10,26, 7,23, 5,22, 3,19, 2,18,33,16,32,13,29,12, 27, 9,25, 8,24, 6,21, 4,20)$ | |
| $ 17, 17 $ | $1$ | $17$ | $( 1,18,34,16,31,13,30,12,28, 9,26, 8,23, 6,22, 4,19)( 2,17,33,15,32,14,29,11, 27,10,25, 7,24, 5,21, 3,20)$ | |
| $ 17, 17 $ | $1$ | $17$ | $( 1,19, 4,22, 6,23, 8,26, 9,28,12,30,13,31,16,34,18)( 2,20, 3,21, 5,24, 7,25, 10,27,11,29,14,32,15,33,17)$ | |
| $ 34 $ | $1$ | $34$ | $( 1,20, 4,21, 6,24, 8,25, 9,27,12,29,13,32,16,33,18, 2,19, 3,22, 5,23, 7,26, 10,28,11,30,14,31,15,34,17)$ | |
| $ 34 $ | $1$ | $34$ | $( 1,21, 8,27,13,33,19, 5,26,11,31,17, 4,24, 9,29,16, 2,22, 7,28,14,34,20, 6, 25,12,32,18, 3,23,10,30,15)$ | |
| $ 17, 17 $ | $1$ | $17$ | $( 1,22, 8,28,13,34,19, 6,26,12,31,18, 4,23, 9,30,16)( 2,21, 7,27,14,33,20, 5, 25,11,32,17, 3,24,10,29,15)$ | |
| $ 17, 17 $ | $1$ | $17$ | $( 1,23,12,34,22, 9,31,19, 8,30,18, 6,28,16, 4,26,13)( 2,24,11,33,21,10,32,20, 7,29,17, 5,27,15, 3,25,14)$ | |
| $ 34 $ | $1$ | $34$ | $( 1,24,12,33,22,10,31,20, 8,29,18, 5,28,15, 4,25,13, 2,23,11,34,21, 9,32,19, 7,30,17, 6,27,16, 3,26,14)$ | |
| $ 34 $ | $1$ | $34$ | $( 1,25,16, 5,30,20, 9,33,23,14, 4,27,18, 7,31,21,12, 2,26,15, 6,29,19,10,34, 24,13, 3,28,17, 8,32,22,11)$ | |
| $ 17, 17 $ | $1$ | $17$ | $( 1,26,16, 6,30,19, 9,34,23,13, 4,28,18, 8,31,22,12)( 2,25,15, 5,29,20,10,33, 24,14, 3,27,17, 7,32,21,11)$ | |
| $ 34 $ | $1$ | $34$ | $( 1,27,19,11, 4,29,22,14, 6,32,23,15, 8,33,26,17, 9, 2,28,20,12, 3,30,21,13, 5,31,24,16, 7,34,25,18,10)$ | |
| $ 17, 17 $ | $1$ | $17$ | $( 1,28,19,12, 4,30,22,13, 6,31,23,16, 8,34,26,18, 9)( 2,27,20,11, 3,29,21,14, 5,32,24,15, 7,33,25,17,10)$ | |
| $ 34 $ | $1$ | $34$ | $( 1,29,23,17,12, 5,34,27,22,15, 9, 3,31,25,19,14, 8, 2,30,24,18,11, 6,33,28, 21,16,10, 4,32,26,20,13, 7)$ | |
| $ 17, 17 $ | $1$ | $17$ | $( 1,30,23,18,12, 6,34,28,22,16, 9, 4,31,26,19,13, 8)( 2,29,24,17,11, 5,33,27, 21,15,10, 3,32,25,20,14, 7)$ | |
| $ 17, 17 $ | $1$ | $17$ | $( 1,31,28,23,19,16,12, 8, 4,34,30,26,22,18,13, 9, 6)( 2,32,27,24,20,15,11, 7, 3,33,29,25,21,17,14,10, 5)$ | |
| $ 34 $ | $1$ | $34$ | $( 1,32,28,24,19,15,12, 7, 4,33,30,25,22,17,13,10, 6, 2,31,27,23,20,16,11, 8, 3,34,29,26,21,18,14, 9, 5)$ | |
| $ 34 $ | $1$ | $34$ | $( 1,33,31,29,28,25,23,21,19,17,16,14,12,10, 8, 5, 4, 2,34,32,30,27,26,24,22, 20,18,15,13,11, 9, 7, 6, 3)$ | |
| $ 17, 17 $ | $1$ | $17$ | $( 1,34,31,30,28,26,23,22,19,18,16,13,12, 9, 8, 6, 4)( 2,33,32,29,27,25,24,21, 20,17,15,14,11,10, 7, 5, 3)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $34=2 \cdot 17$ | magma: Order(G);
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| Cyclic: | yes | magma: IsCyclic(G);
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| Abelian: | yes | magma: IsAbelian(G);
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| Solvable: | yes | magma: IsSolvable(G);
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| Nilpotency class: | $1$ | ||
| Label: | 34.2 | magma: IdentifyGroup(G);
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| Character table: | 34 x 34 character table |
magma: CharacterTable(G);