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Magma
magma: G := TransitiveGroup(14, 7);
Group action invariants
Degree $n$: | $14$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $7$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $F_7 \times C_2$ | ||
CHM label: | $F_{42}(7)[x]2$ | ||
Parity: | $-1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $2$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,9,11)(2,4,8)(3,13,5)(6,12,10), (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), (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 $3$: $C_3$ $4$: $C_2^2$ $6$: $C_6$ x 3 $12$: $C_6\times C_2$ $42$: $F_7$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 7: $F_7$
Low degree siblings
14T7, 28T15, 42T10 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 $ | $1$ | $1$ | $()$ |
$ 6, 6, 1, 1 $ | $7$ | $6$ | $( 2, 4,10,14,12, 6)( 3, 7, 5,13, 9,11)$ |
$ 6, 6, 1, 1 $ | $7$ | $6$ | $( 2, 6,12,14,10, 4)( 3,11, 9,13, 5, 7)$ |
$ 3, 3, 3, 3, 1, 1 $ | $7$ | $3$ | $( 2,10,12)( 3, 5, 9)( 4,14, 6)( 7,13,11)$ |
$ 3, 3, 3, 3, 1, 1 $ | $7$ | $3$ | $( 2,12,10)( 3, 9, 5)( 4, 6,14)( 7,11,13)$ |
$ 2, 2, 2, 2, 2, 2, 1, 1 $ | $7$ | $2$ | $( 2,14)( 3,13)( 4,12)( 5,11)( 6,10)( 7, 9)$ |
$ 2, 2, 2, 2, 2, 2, 2 $ | $7$ | $2$ | $( 1, 2)( 3,14)( 4,13)( 5,12)( 6,11)( 7,10)( 8, 9)$ |
$ 14 $ | $6$ | $14$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14)$ |
$ 6, 6, 2 $ | $7$ | $6$ | $( 1, 2, 5,14,13,10)( 3, 8, 9,12, 7, 6)( 4,11)$ |
$ 6, 6, 2 $ | $7$ | $6$ | $( 1, 2, 7, 4, 3,12)( 5, 8, 9,14,11,10)( 6,13)$ |
$ 6, 6, 2 $ | $7$ | $6$ | $( 1, 2,11, 8, 9, 4)( 3, 6, 5,10,13,12)( 7,14)$ |
$ 6, 6, 2 $ | $7$ | $6$ | $( 1, 2,13, 8, 9, 6)( 3,10)( 4, 7,12,11,14, 5)$ |
$ 7, 7 $ | $6$ | $7$ | $( 1, 3, 5, 7, 9,11,13)( 2, 4, 6, 8,10,12,14)$ |
$ 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 8)( 2, 9)( 3,10)( 4,11)( 5,12)( 6,13)( 7,14)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $84=2^{2} \cdot 3 \cdot 7$ | 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: | yes | magma: IsSolvable(G);
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Nilpotency class: | not nilpotent | ||
Label: | 84.7 | magma: IdentifyGroup(G);
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Character table: |
2 2 2 2 2 2 2 2 1 2 2 2 2 1 2 3 1 1 1 1 1 1 1 . 1 1 1 1 . 1 7 1 . . . . . . 1 . . . . 1 1 1a 6a 6b 3a 3b 2a 2b 14a 6c 6d 6e 6f 7a 2c 2P 1a 3a 3b 3b 3a 1a 1a 7a 3a 3b 3b 3a 7a 1a 3P 1a 2a 2a 1a 1a 2a 2b 14a 2b 2b 2c 2c 7a 2c 5P 1a 6b 6a 3b 3a 2a 2b 14a 6d 6c 6f 6e 7a 2c 7P 1a 6a 6b 3a 3b 2a 2b 2c 6c 6d 6e 6f 1a 2c 11P 1a 6b 6a 3b 3a 2a 2b 14a 6d 6c 6f 6e 7a 2c 13P 1a 6a 6b 3a 3b 2a 2b 14a 6c 6d 6e 6f 7a 2c X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 -1 -1 1 1 -1 -1 1 -1 -1 1 1 1 1 X.3 1 -1 -1 1 1 -1 1 -1 1 1 -1 -1 1 -1 X.4 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 -1 X.5 1 A /A -/A -A -1 -1 1 A /A -/A -A 1 1 X.6 1 /A A -A -/A -1 -1 1 /A A -A -/A 1 1 X.7 1 A /A -/A -A -1 1 -1 -A -/A /A A 1 -1 X.8 1 /A A -A -/A -1 1 -1 -/A -A A /A 1 -1 X.9 1 -/A -A -A -/A 1 -1 -1 /A A A /A 1 -1 X.10 1 -A -/A -/A -A 1 -1 -1 A /A /A A 1 -1 X.11 1 -/A -A -A -/A 1 1 1 -/A -A -A -/A 1 1 X.12 1 -A -/A -/A -A 1 1 1 -A -/A -/A -A 1 1 X.13 6 . . . . . . 1 . . . . -1 -6 X.14 6 . . . . . . -1 . . . . -1 6 A = -E(3) = (1-Sqrt(-3))/2 = -b3 |
magma: CharacterTable(G);