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Magma
magma: G := TransitiveGroup(14, 45);
Group action invariants
Degree $n$: | $14$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $45$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $F_7\wr C_2$ | ||
CHM label: | $[F_{42}(7)^{2}]2=F_{42}(7)wr2$ | ||
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), (2,12)(4,10)(6,8), (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14), (2,4,8)(6,12,10) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $3$: $C_3$ $4$: $C_2^2$ $6$: $S_3$, $C_6$ x 3 $8$: $D_{4}$ $12$: $D_{6}$, $C_6\times C_2$ $18$: $S_3\times C_3$ $24$: $(C_6\times C_2):C_2$, $D_4 \times C_3$ $36$: $C_6\times S_3$ $72$: 12T42 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 7: None
Low degree siblings
28T251, 28T252, 28T253, 42T368, 42T369, 42T370, 42T371, 42T372Siblings 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 $ | $1$ | $1$ | $()$ |
$ 7, 1, 1, 1, 1, 1, 1, 1 $ | $12$ | $7$ | $( 2, 4, 6, 8,10,12,14)$ |
$ 7, 7 $ | $36$ | $7$ | $( 1, 3, 5, 7, 9,11,13)( 2, 4, 6, 8,10,12,14)$ |
$ 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $14$ | $2$ | $( 4,14)( 6,12)( 8,10)$ |
$ 7, 2, 2, 2, 1 $ | $84$ | $14$ | $( 1, 3, 5, 7, 9,11,13)( 4,14)( 6,12)( 8,10)$ |
$ 2, 2, 2, 2, 2, 2, 1, 1 $ | $49$ | $2$ | $( 3,13)( 4,14)( 5,11)( 6,12)( 7, 9)( 8,10)$ |
$ 3, 3, 3, 3, 1, 1 $ | $98$ | $3$ | $( 3, 9, 5)( 4, 6,10)( 7,11,13)( 8,14,12)$ |
$ 6, 3, 3, 1, 1 $ | $98$ | $6$ | $( 3, 9, 5)( 4,12,10,14, 6, 8)( 7,11,13)$ |
$ 6, 3, 3, 1, 1 $ | $98$ | $6$ | $( 3, 7, 5,13, 9,11)( 4, 6,10)( 8,14,12)$ |
$ 6, 6, 1, 1 $ | $98$ | $6$ | $( 3, 7, 5,13, 9,11)( 4,12,10,14, 6, 8)$ |
$ 2, 2, 2, 2, 2, 2, 2 $ | $42$ | $2$ | $( 1, 8)( 2, 9)( 3,10)( 4,11)( 5,12)( 6,13)( 7,14)$ |
$ 14 $ | $252$ | $14$ | $( 1,10, 3,12, 5,14, 7, 2, 9, 4,11, 6,13, 8)$ |
$ 4, 4, 4, 2 $ | $294$ | $4$ | $( 1,10, 3, 8)( 2, 9)( 4,11,14, 7)( 5, 6,13,12)$ |
$ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1 $ | $14$ | $3$ | $( 3, 5, 9)( 7,13,11)$ |
$ 7, 3, 3, 1 $ | $84$ | $21$ | $( 2, 4, 6, 8,10,12,14)( 3, 5, 9)( 7,13,11)$ |
$ 3, 3, 2, 2, 2, 1, 1 $ | $98$ | $6$ | $( 3, 5, 9)( 4,14)( 6,12)( 7,13,11)( 8,10)$ |
$ 6, 1, 1, 1, 1, 1, 1, 1, 1 $ | $14$ | $6$ | $( 3,11, 9,13, 5, 7)$ |
$ 7, 6, 1 $ | $84$ | $42$ | $( 2, 4, 6, 8,10,12,14)( 3,11, 9,13, 5, 7)$ |
$ 6, 2, 2, 2, 1, 1 $ | $98$ | $6$ | $( 3,11, 9,13, 5, 7)( 4,14)( 6,12)( 8,10)$ |
$ 3, 3, 3, 3, 1, 1 $ | $49$ | $3$ | $( 3, 9, 5)( 4,10, 6)( 7,11,13)( 8,12,14)$ |
$ 6, 3, 3, 1, 1 $ | $98$ | $6$ | $( 3, 9, 5)( 4, 8, 6,14,10,12)( 7,11,13)$ |
$ 6, 6, 1, 1 $ | $49$ | $6$ | $( 3, 7, 5,13, 9,11)( 4, 8, 6,14,10,12)$ |
$ 6, 6, 2 $ | $294$ | $6$ | $( 1, 8)( 2, 9,10, 3,12, 5)( 4,11,14, 7, 6,13)$ |
$ 12, 2 $ | $294$ | $12$ | $( 1,10, 3, 6,13,14, 7,12, 5, 2, 9, 8)( 4,11)$ |
$ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1 $ | $14$ | $3$ | $( 3, 9, 5)( 7,11,13)$ |
$ 7, 3, 3, 1 $ | $84$ | $21$ | $( 2, 4, 6, 8,10,12,14)( 3, 9, 5)( 7,11,13)$ |
$ 3, 3, 2, 2, 2, 1, 1 $ | $98$ | $6$ | $( 3, 9, 5)( 4,14)( 6,12)( 7,11,13)( 8,10)$ |
$ 6, 1, 1, 1, 1, 1, 1, 1, 1 $ | $14$ | $6$ | $( 3, 7, 5,13, 9,11)$ |
$ 7, 6, 1 $ | $84$ | $42$ | $( 2, 4, 6, 8,10,12,14)( 3, 7, 5,13, 9,11)$ |
$ 6, 2, 2, 2, 1, 1 $ | $98$ | $6$ | $( 3, 7, 5,13, 9,11)( 4,14)( 6,12)( 8,10)$ |
$ 3, 3, 3, 3, 1, 1 $ | $49$ | $3$ | $( 3, 5, 9)( 4, 6,10)( 7,13,11)( 8,14,12)$ |
$ 6, 3, 3, 1, 1 $ | $98$ | $6$ | $( 3, 5, 9)( 4,12,10,14, 6, 8)( 7,13,11)$ |
$ 6, 6, 1, 1 $ | $49$ | $6$ | $( 3,11, 9,13, 5, 7)( 4,12,10,14, 6, 8)$ |
$ 6, 6, 2 $ | $294$ | $6$ | $( 1, 8)( 2, 9,12, 5,10, 3)( 4,11, 6,13,14, 7)$ |
$ 12, 2 $ | $294$ | $12$ | $( 1,10, 3, 2, 9, 6,13, 4,11,12, 5, 8)( 7,14)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $3528=2^{3} \cdot 3^{2} \cdot 7^{2}$ | 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: | 3528.br | magma: IdentifyGroup(G);
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Character table: not available. |
magma: CharacterTable(G);