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Magma
magma: G := TransitiveGroup(14, 13);
Group action invariants
| Degree $n$: | $14$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $13$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $D_7^2$ | ||
| CHM label: | $[1/2.D(7)^{2}]2$ | ||
| 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: | (2,4,6,8,10,12,14), (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14), (1,13)(2,12)(3,11)(4,10)(5,9)(6,8) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_2^2$ $14$: $D_{7}$ x 2 $28$: $D_{14}$ x 2 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 7: None
Low degree siblings
14T13 x 2, 28T36 x 3Siblings 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, 1, 1 $ | $49$ | $2$ | $( 3,13)( 4,14)( 5,11)( 6,12)( 7, 9)( 8,10)$ |
| $ 7, 1, 1, 1, 1, 1, 1, 1 $ | $4$ | $7$ | $( 2, 4, 6, 8,10,12,14)$ |
| $ 7, 1, 1, 1, 1, 1, 1, 1 $ | $4$ | $7$ | $( 2, 6,10,14, 4, 8,12)$ |
| $ 7, 1, 1, 1, 1, 1, 1, 1 $ | $4$ | $7$ | $( 2, 8,14, 6,12, 4,10)$ |
| $ 2, 2, 2, 2, 2, 2, 2 $ | $7$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)$ |
| $ 2, 2, 2, 2, 2, 2, 2 $ | $7$ | $2$ | $( 1, 2)( 3,14)( 4,13)( 5,12)( 6,11)( 7,10)( 8, 9)$ |
| $ 14 $ | $14$ | $14$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14)$ |
| $ 14 $ | $14$ | $14$ | $( 1, 2, 3,14, 5,12, 7,10, 9, 8,11, 6,13, 4)$ |
| $ 14 $ | $14$ | $14$ | $( 1, 2, 5, 6, 9,10,13,14, 3, 4, 7, 8,11,12)$ |
| $ 14 $ | $14$ | $14$ | $( 1, 2, 5,12, 9, 8,13, 4, 3,14, 7,10,11, 6)$ |
| $ 14 $ | $14$ | $14$ | $( 1, 2, 7, 8,13,14, 5, 6,11,12, 3, 4, 9,10)$ |
| $ 14 $ | $14$ | $14$ | $( 1, 2, 7,10,13, 4, 5,12,11, 6, 3,14, 9, 8)$ |
| $ 7, 7 $ | $2$ | $7$ | $( 1, 3, 5, 7, 9,11,13)( 2, 4, 6, 8,10,12,14)$ |
| $ 7, 7 $ | $4$ | $7$ | $( 1, 3, 5, 7, 9,11,13)( 2, 6,10,14, 4, 8,12)$ |
| $ 7, 7 $ | $4$ | $7$ | $( 1, 3, 5, 7, 9,11,13)( 2, 8,14, 6,12, 4,10)$ |
| $ 7, 7 $ | $4$ | $7$ | $( 1, 3, 5, 7, 9,11,13)( 2,10, 4,12, 6,14, 8)$ |
| $ 7, 7 $ | $4$ | $7$ | $( 1, 3, 5, 7, 9,11,13)( 2,12, 8, 4,14,10, 6)$ |
| $ 7, 7 $ | $2$ | $7$ | $( 1, 3, 5, 7, 9,11,13)( 2,14,12,10, 8, 6, 4)$ |
| $ 7, 7 $ | $2$ | $7$ | $( 1, 5, 9,13, 3, 7,11)( 2, 6,10,14, 4, 8,12)$ |
| $ 7, 7 $ | $4$ | $7$ | $( 1, 5, 9,13, 3, 7,11)( 2, 8,14, 6,12, 4,10)$ |
| $ 7, 7 $ | $4$ | $7$ | $( 1, 5, 9,13, 3, 7,11)( 2,10, 4,12, 6,14, 8)$ |
| $ 7, 7 $ | $2$ | $7$ | $( 1, 5, 9,13, 3, 7,11)( 2,12, 8, 4,14,10, 6)$ |
| $ 7, 7 $ | $2$ | $7$ | $( 1, 7,13, 5,11, 3, 9)( 2, 8,14, 6,12, 4,10)$ |
| $ 7, 7 $ | $2$ | $7$ | $( 1, 7,13, 5,11, 3, 9)( 2,10, 4,12, 6,14, 8)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $196=2^{2} \cdot 7^{2}$ | 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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| Nilpotency class: | not nilpotent | ||
| Label: | 196.9 | magma: IdentifyGroup(G);
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| Character table: not available. |
magma: CharacterTable(G);