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Magma
magma: G := TransitiveGroup(14, 49);
Group action invariants
Degree $n$: | $14$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $49$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $S_7\times C_2$ | ||
CHM label: | $2[x]S(7)$ | ||
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: | (3,5)(10,12), (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14), (1,3,5,7,9,11,13)(2,4,6,8,10,12,14) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_2^2$ $5040$: $S_7$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 7: $S_7$
Low degree siblings
14T49, 28T363, 42T549 x 2, 42T550 x 2Siblings 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 $ | $1$ | $1$ | $()$ |
$ 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 8)( 2, 9)( 3,10)( 4,11)( 5,12)( 6,13)( 7,14)$ |
$ 7, 7 $ | $720$ | $7$ | $( 1, 9,11, 7, 3, 5,13)( 2, 4,14,10,12, 6, 8)$ |
$ 14 $ | $720$ | $14$ | $( 1, 2,11,14, 3,12,13, 8, 9, 4, 7,10, 5, 6)$ |
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $105$ | $2$ | $( 1,13)( 4,14)( 6, 8)( 7,11)$ |
$ 2, 2, 2, 2, 2, 2, 2 $ | $105$ | $2$ | $( 1, 6)( 2, 9)( 3,10)( 4, 7)( 5,12)( 8,13)(11,14)$ |
$ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1 $ | $70$ | $3$ | $( 2,12,10)( 3, 9, 5)$ |
$ 6, 2, 2, 2, 2 $ | $70$ | $6$ | $( 1, 8)( 2, 5,10, 9,12, 3)( 4,11)( 6,13)( 7,14)$ |
$ 4, 4, 1, 1, 1, 1, 1, 1 $ | $210$ | $4$ | $( 1,11,13, 7)( 4, 6,14, 8)$ |
$ 4, 4, 2, 2, 2 $ | $210$ | $4$ | $( 1, 4,13,14)( 2, 9)( 3,10)( 5,12)( 6, 7, 8,11)$ |
$ 3, 3, 2, 2, 2, 2 $ | $210$ | $6$ | $( 1,13)( 2,10,12)( 3, 5, 9)( 4,14)( 6, 8)( 7,11)$ |
$ 6, 2, 2, 2, 2 $ | $210$ | $6$ | $( 1, 6)( 2, 3,12, 9,10, 5)( 4, 7)( 8,13)(11,14)$ |
$ 4, 4, 3, 3 $ | $420$ | $12$ | $( 1, 7,13,11)( 2,12,10)( 3, 9, 5)( 4, 8,14, 6)$ |
$ 6, 4, 4 $ | $420$ | $12$ | $( 1,14,13, 4)( 2, 5,10, 9,12, 3)( 6,11, 8, 7)$ |
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $21$ | $2$ | $( 1,13)( 6, 8)$ |
$ 2, 2, 2, 2, 2, 2, 2 $ | $21$ | $2$ | $( 1, 6)( 2, 9)( 3,10)( 4,11)( 5,12)( 7,14)( 8,13)$ |
$ 3, 3, 2, 2, 1, 1, 1, 1 $ | $420$ | $6$ | $( 1,13)( 2,12,10)( 3, 9, 5)( 6, 8)$ |
$ 6, 2, 2, 2, 2 $ | $420$ | $6$ | $( 1, 6)( 2, 5,10, 9,12, 3)( 4,11)( 7,14)( 8,13)$ |
$ 2, 2, 2, 2, 2, 2, 1, 1 $ | $105$ | $2$ | $( 1,13)( 3,11)( 4,10)( 5, 7)( 6, 8)(12,14)$ |
$ 2, 2, 2, 2, 2, 2, 2 $ | $105$ | $2$ | $( 1, 6)( 2, 9)( 3, 4)( 5,14)( 7,12)( 8,13)(10,11)$ |
$ 3, 3, 3, 3, 1, 1 $ | $280$ | $3$ | $( 1, 5, 3)( 4, 6,14)( 7,11,13)( 8,12,10)$ |
$ 6, 6, 2 $ | $280$ | $6$ | $( 1,12, 3, 8, 5,10)( 2, 9)( 4,13,14,11, 6, 7)$ |
$ 6, 6, 1, 1 $ | $840$ | $6$ | $( 1,11, 5,13, 3, 7)( 4,12, 6,10,14, 8)$ |
$ 6, 6, 2 $ | $840$ | $6$ | $( 1, 4, 5, 6, 3,14)( 2, 9)( 7, 8,11,12,13,10)$ |
$ 5, 5, 1, 1, 1, 1 $ | $504$ | $5$ | $( 1, 9,11, 3, 5)( 2, 4,10,12, 8)$ |
$ 10, 2, 2 $ | $504$ | $10$ | $( 1, 2,11,10, 5, 8, 9, 4, 3,12)( 6,13)( 7,14)$ |
$ 5, 5, 2, 2 $ | $504$ | $10$ | $( 1, 3, 9, 5,11)( 2,12, 4, 8,10)( 6,14)( 7,13)$ |
$ 10, 2, 2 $ | $504$ | $10$ | $( 1,10, 9,12,11, 8, 3, 2, 5, 4)( 6, 7)(13,14)$ |
$ 4, 4, 2, 2, 1, 1 $ | $630$ | $4$ | $( 1, 3)( 2,12,14, 6)( 5, 7,13, 9)( 8,10)$ |
$ 4, 4, 2, 2, 2 $ | $630$ | $4$ | $( 1,10)( 2, 5,14,13)( 3, 8)( 4,11)( 6, 9,12, 7)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $10080=2^{5} \cdot 3^{2} \cdot 5 \cdot 7$ | 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: | 10080.l | magma: IdentifyGroup(G);
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Character table: not available. |
magma: CharacterTable(G);