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Magma
magma: G := TransitiveGroup(35, 24);
Group action invariants
| Degree $n$: | $35$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $24$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $D_5\times \GL(3,2)$ | ||
| 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: | (1,34,7,25,3,31,9,22,5,33,6,24,2,35,8,21,4,32,10,23)(11,29,12,30,13,26,14,27,15,28)(16,19,17,20,18), (1,26,21)(2,30,22,5,27,25)(3,29,23,4,28,24)(6,16,11)(7,20,12,10,17,15)(8,19,13,9,18,14)(32,35)(33,34) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $10$: $D_{5}$ $168$: $\GL(3,2)$ $336$: 14T17 Resolvents shown for degrees $\leq 47$
Subfields
Degree 5: $D_{5}$
Degree 7: $\GL(3,2)$
Low degree siblings
35T24, 40T1570Siblings are shown with degree $\leq 47$
A number field with this Galois group has exactly one arithmetically equivalent field.
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 $ | $1$ | $1$ | $()$ |
| $ 5, 5, 5, 5, 5, 5, 5 $ | $2$ | $5$ | $( 1, 4, 2, 5, 3)( 6, 9, 7,10, 8)(11,14,12,15,13)(16,19,17,20,18) (21,24,22,25,23)(26,29,27,30,28)(31,34,32,35,33)$ |
| $ 5, 5, 5, 5, 5, 5, 5 $ | $2$ | $5$ | $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20) (21,22,23,24,25)(26,27,28,29,30)(31,32,33,34,35)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1 $ | $5$ | $2$ | $( 2, 5)( 3, 4)( 7,10)( 8, 9)(12,15)(13,14)(17,20)(18,19)(22,25)(23,24)(27,30) (28,29)(32,35)(33,34)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $21$ | $2$ | $( 6,26)( 7,27)( 8,28)( 9,29)(10,30)(16,21)(17,22)(18,23)(19,24)(20,25)$ |
| $ 10, 10, 5, 5, 5 $ | $42$ | $10$ | $( 1, 4, 2, 5, 3)( 6,29, 7,30, 8,26, 9,27,10,28)(11,14,12,15,13) (16,24,17,25,18,21,19,22,20,23)(31,34,32,35,33)$ |
| $ 10, 10, 5, 5, 5 $ | $42$ | $10$ | $( 1, 2, 3, 4, 5)( 6,27, 8,29,10,26, 7,28, 9,30)(11,12,13,14,15) (16,22,18,24,20,21,17,23,19,25)(31,32,33,34,35)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1 $ | $105$ | $2$ | $( 2, 5)( 3, 4)( 6,26)( 7,30)( 8,29)( 9,28)(10,27)(12,15)(13,14)(16,21)(17,25) (18,24)(19,23)(20,22)(32,35)(33,34)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1 $ | $56$ | $3$ | $( 6,31,26)( 7,32,27)( 8,33,28)( 9,34,29)(10,35,30)(11,21,16)(12,22,17) (13,23,18)(14,24,19)(15,25,20)$ |
| $ 15, 15, 5 $ | $112$ | $15$ | $( 1, 4, 2, 5, 3)( 6,34,27,10,33,26, 9,32,30, 8,31,29, 7,35,28)(11,24,17,15,23, 16,14,22,20,13,21,19,12,25,18)$ |
| $ 15, 15, 5 $ | $112$ | $15$ | $( 1, 2, 3, 4, 5)( 6,32,28, 9,35,26, 7,33,29,10,31,27, 8,34,30)(11,22,18,14,25, 16,12,23,19,15,21,17,13,24,20)$ |
| $ 6, 6, 6, 6, 3, 3, 2, 2, 1 $ | $280$ | $6$ | $( 2, 5)( 3, 4)( 6,31,26)( 7,35,27,10,32,30)( 8,34,28, 9,33,29)(11,21,16) (12,25,17,15,22,20)(13,24,18,14,23,19)$ |
| $ 20, 10, 5 $ | $84$ | $20$ | $( 1, 5, 4, 3, 2)( 6,20, 9,18, 7,16,10,19, 8,17)(11,25,34,28,12,21,35,29,13,22, 31,30,14,23,32,26,15,24,33,27)$ |
| $ 20, 10, 5 $ | $84$ | $20$ | $( 1, 3, 5, 2, 4)( 6,18,10,17, 9,16, 8,20, 7,19)(11,23,35,27,14,21,33,30,12,24, 31,28,15,22,34,26,13,25,32,29)$ |
| $ 4, 4, 4, 4, 4, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ | $42$ | $4$ | $( 6,16)( 7,17)( 8,18)( 9,19)(10,20)(11,21,31,26)(12,22,32,27)(13,23,33,28) (14,24,34,29)(15,25,35,30)$ |
| $ 4, 4, 4, 4, 4, 2, 2, 2, 2, 2, 2, 2, 1 $ | $210$ | $4$ | $( 1, 2)( 3, 5)( 6,17)( 7,16)( 8,20)( 9,19)(10,18)(11,22,31,27)(12,21,32,26) (13,25,33,30)(14,24,34,29)(15,23,35,28)$ |
| $ 35 $ | $48$ | $35$ | $( 1,34,27,15, 8,16,24, 2,35,28,11, 9,17,25, 3,31,29,12,10,18,21, 4,32,30,13, 6,19,22, 5,33,26,14, 7,20,23)$ |
| $ 35 $ | $48$ | $35$ | $( 1,32,28,14,10,16,22, 3,34,30,11, 7,18,24, 5,31,27,13, 9,20,21, 2,33,29,15, 6,17,23, 4,35,26,12, 8,19,25)$ |
| $ 7, 7, 7, 7, 7 $ | $24$ | $7$ | $( 1,31,26,11, 6,16,21)( 2,32,27,12, 7,17,22)( 3,33,28,13, 8,18,23) ( 4,34,29,14, 9,19,24)( 5,35,30,15,10,20,25)$ |
| $ 14, 14, 7 $ | $120$ | $14$ | $( 1,33,26,13, 6,18,21, 3,31,28,11, 8,16,23)( 2,32,27,12, 7,17,22) ( 4,35,29,15, 9,20,24, 5,34,30,14,10,19,25)$ |
| $ 35 $ | $48$ | $35$ | $( 1,33,30,22,19,11, 8, 5,32,29,21,18,15, 7, 4,31,28,25,17,14, 6, 3,35,27,24, 16,13,10, 2,34,26,23,20,12, 9)$ |
| $ 7, 7, 7, 7, 7 $ | $24$ | $7$ | $( 1,31,26,21,16,11, 6)( 2,32,27,22,17,12, 7)( 3,33,28,23,18,13, 8) ( 4,34,29,24,19,14, 9)( 5,35,30,25,20,15,10)$ |
| $ 35 $ | $48$ | $35$ | $( 1,35,29,23,17,11,10, 4,33,27,21,20,14, 8, 2,31,30,24,18,12, 6, 5,34,28,22, 16,15, 9, 3,32,26,25,19,13, 7)$ |
| $ 14, 14, 7 $ | $120$ | $14$ | $( 1,34,26,24,16,14, 6, 4,31,29,21,19,11, 9)( 2,33,27,23,17,13, 7, 3,32,28,22, 18,12, 8)( 5,35,30,25,20,15,10)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $1680=2^{4} \cdot 3 \cdot 5 \cdot 7$ | 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: | 1680.926 | magma: IdentifyGroup(G);
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| Character table: not available. |
magma: CharacterTable(G);