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Magma
magma: G := TransitiveGroup(34, 9);
Group action invariants
| Degree $n$: | $34$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $9$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_2\times F_{17}$ | ||
| Parity: | $-1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
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| Nilpotency class: | $-1$ (not nilpotent) | magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $2$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,20,9,8,27,32,26,17,30,12,22,23,4,33,5,13)(2,19,10,7,28,31,25,18,29,11,21,24,3,34,6,14)(15,16), (1,4,24,19,14,22,34,18,27,26,5,9,15,7,30,11)(2,3,23,20,13,21,33,17,28,25,6,10,16,8,29,12) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_4$ x 2, $C_2^2$ $8$: $C_8$ x 2, $C_4\times C_2$ $16$: $C_{16}$ x 2, $C_8\times C_2$ $32$: 32T32 $272$: $F_{17}$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 17: $F_{17}$
Low degree siblings
34T9Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
| 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, 17 $ | $16$ | $17$ | $( 1,34,31,30,27,26,24,22,19,18,15,14,11, 9, 7, 5, 4)( 2,33,32,29,28,25,23,21, 20,17,16,13,12,10, 8, 6, 3)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $17$ | $2$ | $( 3,33)( 4,34)( 5,31)( 6,32)( 7,30)( 8,29)( 9,27)(10,28)(11,26)(12,25)(13,23) (14,24)(15,22)(16,21)(17,20)(18,19)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 1, 1 $ | $17$ | $4$ | $( 3,10,33,28)( 4, 9,34,27)( 5,18,31,19)( 6,17,32,20)( 7,26,30,11)( 8,25,29,12) (13,16,23,21)(14,15,24,22)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 1, 1 $ | $17$ | $4$ | $( 3,28,33,10)( 4,27,34, 9)( 5,19,31,18)( 6,20,32,17)( 7,11,30,26)( 8,12,29,25) (13,21,23,16)(14,22,24,15)$ |
| $ 8, 8, 8, 8, 1, 1 $ | $17$ | $8$ | $( 3,32,10,20,33, 6,28,17)( 4,31, 9,19,34, 5,27,18)( 7,24,26,22,30,14,11,15) ( 8,23,25,21,29,13,12,16)$ |
| $ 8, 8, 8, 8, 1, 1 $ | $17$ | $8$ | $( 3, 6,10,17,33,32,28,20)( 4, 5, 9,18,34,31,27,19)( 7,14,26,15,30,24,11,22) ( 8,13,25,16,29,23,12,21)$ |
| $ 8, 8, 8, 8, 1, 1 $ | $17$ | $8$ | $( 3,20,28,32,33,17,10, 6)( 4,19,27,31,34,18, 9, 5)( 7,22,11,24,30,15,26,14) ( 8,21,12,23,29,16,25,13)$ |
| $ 8, 8, 8, 8, 1, 1 $ | $17$ | $8$ | $( 3,17,28, 6,33,20,10,32)( 4,18,27, 5,34,19, 9,31)( 7,15,11,14,30,22,26,24) ( 8,16,12,13,29,21,25,23)$ |
| $ 16, 16, 2 $ | $17$ | $16$ | $( 1,20, 9, 8,27,32,26,17,30,12,22,23, 4,33, 5,13)( 2,19,10, 7,28,31,25,18,29, 11,21,24, 3,34, 6,14)(15,16)$ |
| $ 16, 16, 2 $ | $17$ | $16$ | $( 1,17, 7,10,30,25,19,28, 5,23,34,32,11,16,22,13)( 2,18, 8, 9,29,26,20,27, 6, 24,33,31,12,15,21,14)( 3, 4)$ |
| $ 16, 16, 2 $ | $17$ | $16$ | $( 1, 6,15,23, 9,25,31,29, 7, 3,27,20,34,17,11,13)( 2, 5,16,24,10,26,32,30, 8, 4,28,19,33,18,12,14)(21,22)$ |
| $ 16, 16, 2 $ | $17$ | $16$ | $( 1,32, 7,33,19, 3, 9,12,24,28,18,25, 5,21,15,13)( 2,31, 8,34,20, 4,10,11,23, 27,17,26, 6,22,16,14)(29,30)$ |
| $ 16, 16, 2 $ | $17$ | $16$ | $( 1,33,27,10,24,32,22,25, 4, 6,11,29,15, 8,18,13)( 2,34,28, 9,23,31,21,26, 3, 5,12,30,16, 7,17,14)(19,20)$ |
| $ 16, 16, 2 $ | $17$ | $16$ | $( 1, 3,31,16,30,21,11, 8,19,17,24, 6,26,33, 9,13)( 2, 4,32,15,29,22,12, 7,20, 18,23, 5,25,34,10,14)(27,28)$ |
| $ 16, 16, 2 $ | $17$ | $16$ | $( 1,28,34, 3,18,16,26,10,22,29,24,20, 5, 8,31,13)( 2,27,33, 4,17,15,25, 9,21, 30,23,19, 6, 7,32,14)(11,12)$ |
| $ 16, 16, 2 $ | $17$ | $16$ | $( 1,10,15,12,26,28, 4,20,31,23,18,21, 7, 6,30,13)( 2, 9,16,11,25,27, 3,19,32, 24,17,22, 8, 5,29,14)(33,34)$ |
| $ 16, 16, 1, 1 $ | $17$ | $16$ | $( 3,21,32,29,10,13,20,12,33,16, 6, 8,28,23,17,25)( 4,22,31,30, 9,14,19,11,34, 15, 5, 7,27,24,18,26)$ |
| $ 16, 16, 1, 1 $ | $17$ | $16$ | $( 3,16,32, 8,10,23,20,25,33,21, 6,29,28,13,17,12)( 4,15,31, 7, 9,24,19,26,34, 22, 5,30,27,14,18,11)$ |
| $ 16, 16, 1, 1 $ | $17$ | $16$ | $( 3,13, 6,25,10,16,17,29,33,23,32,12,28,21,20, 8)( 4,14, 5,26, 9,15,18,30,34, 24,31,11,27,22,19, 7)$ |
| $ 16, 16, 1, 1 $ | $17$ | $16$ | $( 3,23, 6,12,10,21,17, 8,33,13,32,25,28,16,20,29)( 4,24, 5,11, 9,22,18, 7,34, 14,31,26,27,15,19,30)$ |
| $ 16, 16, 1, 1 $ | $17$ | $16$ | $( 3,29,20,16,28,25,32,13,33, 8,17,21,10,12, 6,23)( 4,30,19,15,27,26,31,14,34, 7,18,22, 9,11, 5,24)$ |
| $ 16, 16, 1, 1 $ | $17$ | $16$ | $( 3, 8,20,21,28,12,32,23,33,29,17,16,10,25, 6,13)( 4, 7,19,22,27,11,31,24,34, 30,18,15, 9,26, 5,14)$ |
| $ 16, 16, 1, 1 $ | $17$ | $16$ | $( 3,12,17,13,28,29, 6,21,33,25,20,23,10, 8,32,16)( 4,11,18,14,27,30, 5,22,34, 26,19,24, 9, 7,31,15)$ |
| $ 16, 16, 1, 1 $ | $17$ | $16$ | $( 3,25,17,23,28, 8, 6,16,33,12,20,13,10,29,32,21)( 4,26,18,24,27, 7, 5,15,34, 11,19,14, 9,30,31,22)$ |
| $ 8, 8, 8, 8, 2 $ | $17$ | $8$ | $( 1,20,22,25,34,16,14,10)( 2,19,21,26,33,15,13, 9)( 3,24,29, 7,32,11, 6,27) ( 4,23,30, 8,31,12, 5,28)(17,18)$ |
| $ 8, 8, 8, 8, 2 $ | $17$ | $8$ | $( 1,17,19,16,24, 8, 5,10)( 2,18,20,15,23, 7, 6, 9)( 3,14,28,34,21,11,32,26) ( 4,13,27,33,22,12,31,25)(29,30)$ |
| $ 8, 8, 8, 8, 2 $ | $17$ | $8$ | $( 1, 6, 4,21,30,25,27,10)( 2, 5, 3,22,29,26,28, 9)( 7,20,14,33,24,12,18,32) ( 8,19,13,34,23,11,17,31)(15,16)$ |
| $ 8, 8, 8, 8, 2 $ | $17$ | $8$ | $( 1,32,30,12,19,23,26,10)( 2,31,29,11,20,24,25, 9)( 3,15,21, 7,17, 5,33,14) ( 4,16,22, 8,18, 6,34,13)(27,28)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 2 $ | $17$ | $4$ | $( 1,33, 7,10)( 2,34, 8, 9)( 3,26, 6,18)( 4,25, 5,17)(11,28,31,16)(12,27,32,15) (13,19,29,24)(14,20,30,23)(21,22)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 2 $ | $17$ | $4$ | $( 1, 3,11,10)( 2, 4,12, 9)( 5,20, 7,28)( 6,19, 8,27)(13,18,33,30)(14,17,34,29) (15,25,31,21)(16,26,32,22)(23,24)$ |
| $ 34 $ | $16$ | $34$ | $( 1,28,19,12, 4,29,22,13, 5,32,24,16, 7,33,26,17, 9, 2,27,20,11, 3,30,21,14, 6,31,23,15, 8,34,25,18,10)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)(29,30)(31,32)(33,34)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $17$ | $2$ | $( 1,10)( 2, 9)( 3, 7)( 4, 8)( 5, 6)(11,33)(12,34)(13,31)(14,32)(15,29)(16,30) (17,27)(18,28)(19,25)(20,26)(21,24)(22,23)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $544=2^{5} \cdot 17$ | 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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| Label: | 544.242 | magma: IdentifyGroup(G);
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| Character table: not available. |
magma: CharacterTable(G);