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Magma
magma: G := TransitiveGroup(34, 43);
Group action invariants
Degree $n$: | $34$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $43$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_2\times \SOMinus(4,4)$ | ||
Parity: | $-1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $2$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,9,12,34,5,31,7,26,4,29,28,21,19,13,16,2,10,11,33,6,32,8,25,3,30,27,22,20,14,15)(17,18)(23,24), (1,21,12,6)(2,22,11,5)(3,32,13,17)(4,31,14,18)(7,27,10,34)(8,28,9,33)(15,24,20,30)(16,23,19,29)(25,26) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_2^2$ $8160$: $\PSL(2,16):C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 17: $\PSL(2,16):C_2$
Low degree siblings
34T43Siblings 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 $ | $1$ | $1$ | $()$ |
$ 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $68$ | $2$ | $( 1,32)( 2,31)( 5,33)( 6,34)( 9,20)(10,19)(13,21)(14,22)(15,26)(16,25)(17,28) (18,27)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $68$ | $2$ | $( 1,31)( 2,32)( 3, 4)( 5,34)( 6,33)( 7, 8)( 9,19)(10,20)(11,12)(13,22)(14,21) (15,25)(16,26)(17,27)(18,28)(23,24)(29,30)$ |
$ 10, 10, 10, 2, 2 $ | $272$ | $10$ | $( 1, 2)( 3,30, 8,12,23, 4,29, 7,11,24)( 5,26,14, 9,28, 6,25,13,10,27) (15,22,20,17,34,16,21,19,18,33)(31,32)$ |
$ 5, 5, 5, 5, 5, 5, 1, 1, 1, 1 $ | $272$ | $5$ | $( 3,29, 8,11,23)( 4,30, 7,12,24)( 5,25,14,10,28)( 6,26,13, 9,27) (15,21,20,18,34)(16,22,19,17,33)$ |
$ 10, 10, 10, 2, 2 $ | $272$ | $10$ | $( 1, 2)( 3,12,29,24, 8, 4,11,30,23, 7)( 5, 9,25,27,14, 6,10,26,28,13) (15,17,21,33,20,16,18,22,34,19)(31,32)$ |
$ 5, 5, 5, 5, 5, 5, 1, 1, 1, 1 $ | $272$ | $5$ | $( 3,11,29,23, 8)( 4,12,30,24, 7)( 5,10,25,28,14)( 6, 9,26,27,13) (15,18,21,34,20)(16,17,22,33,19)$ |
$ 10, 10, 5, 5, 2, 2 $ | $816$ | $10$ | $( 1,32)( 2,31)( 3,11,29,23, 8)( 4,12,30,24, 7)( 5,19,25,17,14,33,10,16,28,22) ( 6,20,26,18,13,34, 9,15,27,21)$ |
$ 10, 10, 10, 2, 2 $ | $816$ | $10$ | $( 1,31)( 2,32)( 3,12,29,24, 8, 4,11,30,23, 7)( 5,20,25,18,14,34,10,15,28,21) ( 6,19,26,17,13,33, 9,16,27,22)$ |
$ 10, 10, 10, 2, 2 $ | $816$ | $10$ | $( 1,31)( 2,32)( 3,30, 8,12,23, 4,29, 7,11,24)( 5,15,14,20,28,34,25,21,10,18) ( 6,16,13,19,27,33,26,22, 9,17)$ |
$ 10, 10, 5, 5, 2, 2 $ | $816$ | $10$ | $( 1,32)( 2,31)( 3,29, 8,11,23)( 4,30, 7,12,24)( 5,16,14,19,28,33,25,22,10,17) ( 6,15,13,20,27,34,26,21, 9,18)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $255$ | $2$ | $( 1,18)( 2,17)( 3,28)( 4,27)( 5,13)( 6,14)( 7,26)( 8,25)( 9,16)(10,15)(11,33) (12,34)(19,20)(21,32)(22,31)(23,30)(24,29)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $255$ | $2$ | $( 1,17)( 2,18)( 3,27)( 4,28)( 5,14)( 6,13)( 7,25)( 8,26)( 9,15)(10,16)(11,34) (12,33)(21,31)(22,32)(23,29)(24,30)$ |
$ 4, 4, 4, 4, 4, 4, 4, 4, 2 $ | $1020$ | $4$ | $( 1,34,17,11)( 2,33,18,12)( 3, 5,27,14)( 4, 6,28,13)( 7,21,25,31)( 8,22,26,32) ( 9,30,15,24)(10,29,16,23)(19,20)$ |
$ 4, 4, 4, 4, 4, 4, 4, 4, 1, 1 $ | $1020$ | $4$ | $( 1,33,17,12)( 2,34,18,11)( 3, 6,27,13)( 4, 5,28,14)( 7,22,25,32)( 8,21,26,31) ( 9,29,15,23)(10,30,16,24)$ |
$ 34 $ | $480$ | $34$ | $( 1,13,16, 9,19,23,30,11,28,18,25,34,32, 8, 5,21, 4, 2,14,15,10,20,24,29,12, 27,17,26,33,31, 7, 6,22, 3)$ |
$ 17, 17 $ | $480$ | $17$ | $( 1,14,16,10,19,24,30,12,28,17,25,33,32, 7, 5,22, 4)( 2,13,15, 9,20,23,29,11, 27,18,26,34,31, 8, 6,21, 3)$ |
$ 34 $ | $480$ | $34$ | $( 1,15,19,29,28,26,32, 6, 4,13,10,23,12,18,33, 8,22, 2,16,20,30,27,25,31, 5, 3,14, 9,24,11,17,34, 7,21)$ |
$ 17, 17 $ | $480$ | $17$ | $( 1,16,19,30,28,25,32, 5, 4,14,10,24,12,17,33, 7,22)( 2,15,20,29,27,26,31, 6, 3,13, 9,23,11,18,34, 8,21)$ |
$ 17, 17 $ | $480$ | $17$ | $( 1,30,32,14,12, 7,16,28, 5,10,17,22,19,25, 4,24,33)( 2,29,31,13,11, 8,15,27, 6, 9,18,21,20,26, 3,23,34)$ |
$ 34 $ | $480$ | $34$ | $( 1,29,32,13,12, 8,16,27, 5, 9,17,21,19,26, 4,23,33, 2,30,31,14,11, 7,15,28, 6,10,18,22,20,25, 3,24,34)$ |
$ 17, 17 $ | $480$ | $17$ | $( 1,32,12,16, 5,17,19, 4,33,30,14, 7,28,10,22,25,24)( 2,31,11,15, 6,18,20, 3, 34,29,13, 8,27, 9,21,26,23)$ |
$ 34 $ | $480$ | $34$ | $( 1,31,12,15, 5,18,19, 3,33,29,14, 8,28, 9,22,26,24, 2,32,11,16, 6,17,20, 4, 34,30,13, 7,27,10,21,25,23)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1 $ | $272$ | $3$ | $( 1,10,14)( 2, 9,13)( 3, 8,11)( 4, 7,12)( 5,16,17)( 6,15,18)(19,22,32) (20,21,31)(25,28,33)(26,27,34)$ |
$ 6, 6, 6, 6, 6, 2, 2 $ | $272$ | $6$ | $( 1, 9,14, 2,10,13)( 3, 7,11, 4, 8,12)( 5,15,17, 6,16,18)(19,21,32,20,22,31) (23,24)(25,27,33,26,28,34)(29,30)$ |
$ 15, 15, 1, 1, 1, 1 $ | $544$ | $15$ | $( 1,22,16, 4,28,10,32,17, 7,33,14,19, 5,12,25)( 2,21,15, 3,27, 9,31,18, 8,34, 13,20, 6,11,26)$ |
$ 30, 2, 2 $ | $544$ | $30$ | $( 1,21,16, 3,28, 9,32,18, 7,34,14,20, 5,11,25, 2,22,15, 4,27,10,31,17, 8,33, 13,19, 6,12,26)(23,24)(29,30)$ |
$ 30, 2, 2 $ | $544$ | $30$ | $( 1,18,25,31,12, 9, 5,27,19, 3,14,15,33,21, 7, 2,17,26,32,11,10, 6,28,20, 4, 13,16,34,22, 8)(23,24)(29,30)$ |
$ 15, 15, 1, 1, 1, 1 $ | $544$ | $15$ | $( 1,17,25,32,12,10, 5,28,19, 4,14,16,33,22, 7)( 2,18,26,31,11, 9, 6,27,20, 3, 13,15,34,21, 8)$ |
$ 6, 6, 6, 6, 6, 2, 2 $ | $1360$ | $6$ | $( 1, 6,19,31,33, 9)( 2, 5,20,32,34,10)( 3,24,11, 4,23,12)( 7, 8) (13,28,26,22,18,16)(14,27,25,21,17,15)(29,30)$ |
$ 6, 6, 6, 6, 3, 3, 1, 1, 1, 1 $ | $1360$ | $6$ | $( 1, 5,19,32,33,10)( 2, 6,20,31,34, 9)( 3,23,11)( 4,24,12)(13,27,26,21,18,15) (14,28,25,22,17,16)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $16320=2^{6} \cdot 3 \cdot 5 \cdot 17$ | 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: | no | magma: IsSolvable(G);
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Nilpotency class: | not nilpotent | ||
Label: | 16320.f | magma: IdentifyGroup(G);
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Character table: not available. |
magma: CharacterTable(G);