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Magma
magma: G := TransitiveGroup(14, 18);
Group action invariants
Degree $n$: | $14$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $18$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $F_8:C_6$ | ||
CHM label: | $[2^{4}]F_{21}(7)$ | ||
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: | (1,9,11)(2,4,8)(3,13,5)(6,12,10), (1,8)(2,9)(4,11), (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) $2$: $C_2$ $3$: $C_3$ $6$: $C_6$ $21$: $C_7:C_3$ $42$: $(C_7:C_3) \times C_2$ $168$: $C_2^3:(C_7: C_3)$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 7: $C_7:C_3$
Low degree siblings
16T712, 28T44, 42T67Siblings 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, 1, 1, 1, 1, 1, 1, 1, 1 $ | $7$ | $2$ | $( 4,11)( 5,12)( 7,14)$ |
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $7$ | $2$ | $( 3,10)( 5,12)( 6,13)( 7,14)$ |
$ 3, 3, 3, 3, 1, 1 $ | $28$ | $3$ | $( 2, 3, 5)( 4, 7,13)( 6,11,14)( 9,10,12)$ |
$ 6, 3, 3, 1, 1 $ | $28$ | $6$ | $( 2, 3, 5, 9,10,12)( 4,14,13)( 6,11, 7)$ |
$ 6, 3, 3, 1, 1 $ | $28$ | $6$ | $( 2, 5,10, 9,12, 3)( 4, 6,14)( 7,11,13)$ |
$ 3, 3, 3, 3, 1, 1 $ | $28$ | $3$ | $( 2, 5, 3)( 4,13, 7)( 6,14,11)( 9,12,10)$ |
$ 14 $ | $24$ | $14$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14)$ |
$ 7, 7 $ | $24$ | $7$ | $( 1, 2, 3, 4,12, 6, 7)( 5,13,14, 8, 9,10,11)$ |
$ 6, 6, 2 $ | $28$ | $6$ | $( 1, 2, 4, 8, 9,11)( 3, 6,12,10,13, 5)( 7,14)$ |
$ 6, 3, 3, 2 $ | $28$ | $6$ | $( 1, 2, 4)( 3,13,12,10, 6, 5)( 7,14)( 8, 9,11)$ |
$ 6, 3, 3, 2 $ | $28$ | $6$ | $( 1, 2, 6, 8, 9,13)( 3,10)( 4, 7, 5)(11,14,12)$ |
$ 6, 6, 2 $ | $28$ | $6$ | $( 1, 2, 6, 8, 9,13)( 3,10)( 4,14, 5,11, 7,12)$ |
$ 14 $ | $24$ | $14$ | $( 1, 4, 7,10,13, 2, 5, 8,11,14, 3, 6, 9,12)$ |
$ 7, 7 $ | $24$ | $7$ | $( 1, 4,14,10,13, 2, 5)( 3, 6, 9,12, 8,11, 7)$ |
$ 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: | $336=2^{4} \cdot 3 \cdot 7$ | 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: | 336.210 | magma: IdentifyGroup(G);
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Character table: |
2 4 4 4 2 2 2 2 1 1 2 2 2 2 1 1 4 3 1 1 1 1 1 1 1 . . 1 1 1 1 . . 1 7 1 . . . . . . 1 1 . . . . 1 1 1 1a 2a 2b 3a 6a 6b 3b 14a 7a 6c 6d 6e 6f 14b 7b 2c 2P 1a 1a 1a 3b 3b 3a 3a 7a 7a 3b 3b 3a 3a 7b 7b 1a 3P 1a 2a 2b 1a 2a 2a 1a 14b 7b 2c 2b 2b 2c 14a 7a 2c 5P 1a 2a 2b 3b 6b 6a 3a 14b 7b 6f 6e 6d 6c 14a 7a 2c 7P 1a 2a 2b 3a 6a 6b 3b 2c 1a 6c 6d 6e 6f 2c 1a 2c 11P 1a 2a 2b 3b 6b 6a 3a 14a 7a 6f 6e 6d 6c 14b 7b 2c 13P 1a 2a 2b 3a 6a 6b 3b 14b 7b 6c 6d 6e 6f 14a 7a 2c X.1 1 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 1 -1 X.3 1 -1 1 A -A -/A /A -1 1 -A A /A -/A -1 1 -1 X.4 1 -1 1 /A -/A -A A -1 1 -/A /A A -A -1 1 -1 X.5 1 1 1 A A /A /A 1 1 A A /A /A 1 1 1 X.6 1 1 1 /A /A A A 1 1 /A /A A A 1 1 1 X.7 3 -3 3 . . . . B -B . . . . /B -/B -3 X.8 3 -3 3 . . . . /B -/B . . . . B -B -3 X.9 3 3 3 . . . . -/B -/B . . . . -B -B 3 X.10 3 3 3 . . . . -B -B . . . . -/B -/B 3 X.11 7 1 -1 1 1 1 1 . . -1 -1 -1 -1 . . -7 X.12 7 -1 -1 1 -1 -1 1 . . 1 -1 -1 1 . . 7 X.13 7 1 -1 A A /A /A . . -A -A -/A -/A . . -7 X.14 7 1 -1 /A /A A A . . -/A -/A -A -A . . -7 X.15 7 -1 -1 A -A -/A /A . . A -A -/A /A . . 7 X.16 7 -1 -1 /A -/A -A A . . /A -/A -A A . . 7 A = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 B = -E(7)-E(7)^2-E(7)^4 = (1-Sqrt(-7))/2 = -b7 |
magma: CharacterTable(G);