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Magma
magma: G := TransitiveGroup(14, 42);
Group action invariants
| Degree $n$: | $14$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $42$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_2^4.\GL(3,2)$ | ||
| CHM label: | $2^{4}`L_{7}(14)$ | ||
| 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), (1,8)(2,11)(3,10)(4,9)(5,6)(12,13), (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) $2$: $C_2$ $168$: $\GL(3,2)$ $336$: 14T17 $1344$: 14T33 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 7: $\GL(3,2)$
Low degree siblings
14T42, 28T229 x 2, 28T230, 28T231 x 2, 42T327 x 2, 42T330 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 $ | $1$ | $1$ | $()$ |
| $ 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 $ | $7$ | $2$ | $( 2, 9)( 4,11)( 5,12)( 6,13)$ |
| $ 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 8)( 2, 9)( 3,10)( 4,11)( 5,12)( 6,13)( 7,14)$ |
| $ 4, 4, 2, 2, 2 $ | $42$ | $4$ | $( 1, 8)( 2, 9)( 3, 5,10,12)( 4,11)( 6, 7,13,14)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1 $ | $84$ | $2$ | $( 1, 8)( 2, 9)( 3,12)( 5,10)( 6,14)( 7,13)$ |
| $ 4, 4, 2, 2, 1, 1 $ | $42$ | $4$ | $( 1, 8)( 3, 5,10,12)( 4,11)( 6,14,13, 7)$ |
| $ 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $84$ | $2$ | $( 1, 8)( 3,12)( 5,10)( 6, 7)(13,14)$ |
| $ 4, 4, 2, 1, 1, 1, 1 $ | $42$ | $4$ | $( 2, 9)( 3,12,10, 5)( 6, 7,13,14)$ |
| $ 4, 4, 1, 1, 1, 1, 1, 1 $ | $42$ | $4$ | $( 3,12,10, 5)( 6,14,13, 7)$ |
| $ 8, 4, 2 $ | $168$ | $8$ | $( 1, 8)( 2,10, 4,14, 9, 3,11, 7)( 5, 6,12,13)$ |
| $ 8, 2, 2, 2 $ | $168$ | $8$ | $( 1, 8)( 2,10,11,14, 9, 3, 4, 7)( 5, 6)(12,13)$ |
| $ 8, 4, 1, 1 $ | $168$ | $8$ | $( 2, 3,11,14, 9,10, 4, 7)( 5, 6,12,13)$ |
| $ 8, 2, 2, 1, 1 $ | $168$ | $8$ | $( 2, 3, 4,14, 9,10,11, 7)( 5, 6)(12,13)$ |
| $ 3, 3, 3, 3, 1, 1 $ | $224$ | $3$ | $( 2, 3,12)( 4, 7, 6)( 5, 9,10)(11,14,13)$ |
| $ 6, 3, 3, 1, 1 $ | $224$ | $6$ | $( 2, 3, 5, 9,10,12)( 4,14,13)( 6,11, 7)$ |
| $ 6, 6, 2 $ | $224$ | $6$ | $( 1, 8)( 2,10, 5, 9, 3,12)( 4,14,13,11, 7, 6)$ |
| $ 6, 3, 3, 2 $ | $224$ | $6$ | $( 1, 8)( 2,10,12)( 3, 5, 9)( 4, 7, 6,11,14,13)$ |
| $ 7, 7 $ | $192$ | $7$ | $( 1, 2, 3, 4,12, 6, 7)( 5,13,14, 8, 9,10,11)$ |
| $ 14 $ | $192$ | $14$ | $( 1, 2, 3,11,12, 6,14, 8, 9,10, 4, 5,13, 7)$ |
| $ 14 $ | $192$ | $14$ | $( 1, 9, 3,14,13,11,12, 8, 2,10, 7, 6, 4, 5)$ |
| $ 7, 7 $ | $192$ | $7$ | $( 1, 9, 3, 7, 6,11, 5)( 2,10,14,13, 4,12, 8)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $2688=2^{7} \cdot 3 \cdot 7$ | 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: | no | magma: IsSolvable(G);
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| Nilpotency class: | not nilpotent | ||
| Label: | 2688.cc | magma: IdentifyGroup(G);
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| Character table: not available. |
magma: CharacterTable(G);