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Magma
magma: G := TransitiveGroup(14, 9);
Group action invariants
| Degree $n$: | $14$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $9$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_2\times F_8$ | ||
| CHM label: | $[2^{4}]7$ | ||
| 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: | (1,8)(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) $2$: $C_2$ $7$: $C_7$ $14$: $C_{14}$ $56$: $C_2^3:C_7$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 7: $C_7$
Low degree siblings
16T196, 28T19 x 3, 28T20Siblings 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 $ | $7$ | $2$ | $( 3,10)( 5,12)( 6,13)( 7,14)$ |
| $ 14 $ | $8$ | $14$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14)$ |
| $ 7, 7 $ | $8$ | $7$ | $( 1, 2, 3, 4,12, 6, 7)( 5,13,14, 8, 9,10,11)$ |
| $ 7, 7 $ | $8$ | $7$ | $( 1, 3, 5, 7, 9,11,13)( 2, 4, 6, 8,10,12,14)$ |
| $ 14 $ | $8$ | $14$ | $( 1, 3, 5,14, 9,11, 6, 8,10,12, 7, 2, 4,13)$ |
| $ 14 $ | $8$ | $14$ | $( 1, 4, 7,10,13, 2, 5, 8,11,14, 3, 6, 9,12)$ |
| $ 7, 7 $ | $8$ | $7$ | $( 1, 4,14,10,13, 2, 5)( 3, 6, 9,12, 8,11, 7)$ |
| $ 14 $ | $8$ | $14$ | $( 1, 5, 2, 6,10,14,11, 8,12, 9,13, 3, 7, 4)$ |
| $ 7, 7 $ | $8$ | $7$ | $( 1, 5, 9,13, 3, 7,11)( 2, 6,10,14, 4, 8,12)$ |
| $ 7, 7 $ | $8$ | $7$ | $( 1, 6, 4, 2, 7,12, 3)( 5,10, 8,13,11, 9,14)$ |
| $ 14 $ | $8$ | $14$ | $( 1, 6, 4, 9,14,12,10, 8,13,11, 2, 7, 5, 3)$ |
| $ 7, 7 $ | $8$ | $7$ | $( 1, 7, 6,12, 4, 3, 2)( 5,11,10, 9, 8,14,13)$ |
| $ 14 $ | $8$ | $14$ | $( 1, 7,13, 5, 4,10, 9, 8,14, 6,12,11, 3, 2)$ |
| $ 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: | $112=2^{4} \cdot 7$ | 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: | 112.41 | magma: IdentifyGroup(G);
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| Character table: |
2 4 4 4 1 1 1 1 1 1 1 1 1 1 1 1 4
7 1 . . 1 1 1 1 1 1 1 1 1 1 1 1 1
1a 2a 2b 14a 7a 7b 14b 14c 7c 14d 7d 7e 14e 7f 14f 2c
2P 1a 1a 1a 7b 7b 7d 7d 7f 7f 7a 7a 7c 7c 7e 7e 1a
3P 1a 2a 2b 14c 7c 7f 14f 14b 7b 14e 7e 7a 14a 7d 14d 2c
5P 1a 2a 2b 14e 7e 7c 14c 14a 7a 14f 7f 7d 14d 7b 14b 2c
7P 1a 2a 2b 2c 1a 1a 2c 2c 1a 2c 1a 1a 2c 1a 2c 2c
11P 1a 2a 2b 14d 7d 7a 14a 14e 7e 14b 7b 7f 14f 7c 14c 2c
13P 1a 2a 2b 14f 7f 7e 14e 14d 7d 14c 7c 7b 14b 7a 14a 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 -B B C -C /C -/C -/B /B -/A /A -1
X.4 1 -1 1 B -B -/C /C /A -/A A -A -C C -/B /B -1
X.5 1 -1 1 C -C -/A /A B -B /B -/B -A A -/C /C -1
X.6 1 -1 1 /C -/C -A A /B -/B B -B -/A /A -C C -1
X.7 1 -1 1 /B -/B -C C A -A /A -/A -/C /C -B B -1
X.8 1 -1 1 /A -/A -/B /B /C -/C C -C -B B -A A -1
X.9 1 1 1 -/A -/A -/B -/B -/C -/C -C -C -B -B -A -A 1
X.10 1 1 1 -/B -/B -C -C -A -A -/A -/A -/C -/C -B -B 1
X.11 1 1 1 -/C -/C -A -A -/B -/B -B -B -/A -/A -C -C 1
X.12 1 1 1 -C -C -/A -/A -B -B -/B -/B -A -A -/C -/C 1
X.13 1 1 1 -B -B -/C -/C -/A -/A -A -A -C -C -/B -/B 1
X.14 1 1 1 -A -A -B -B -C -C -/C -/C -/B -/B -/A -/A 1
X.15 7 1 -1 . . . . . . . . . . . . -7
X.16 7 -1 -1 . . . . . . . . . . . . 7
A = -E(7)
B = -E(7)^2
C = -E(7)^3
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magma: CharacterTable(G);