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Magma
magma: G := TransitiveGroup(14, 40);
Group action invariants
Degree $n$: | $14$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $40$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_2^6:F_7$ | ||
CHM label: | $1/2[2^{7}]F_{42}(7)$ | ||
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), (1,13)(2,12)(3,11)(4,10)(5,9)(6,8)(7,14), (2,9)(7,14), (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) $2$: $C_2$ $3$: $C_3$ $6$: $C_6$ $42$: $F_7$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 7: $F_7$
Low degree siblings
14T41, 16T1502, 28T215, 28T227, 28T228, 28T237, 42T314, 42T315, 42T316, 42T317, 42T318, 42T319Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
Label | Cycle Type | Size | Order | Representative |
$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ | |
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $21$ | $2$ | $(1,8)(2,9)$ | |
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $21$ | $2$ | $( 1, 8)( 2, 9)( 5,12)( 6,13)$ | |
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $14$ | $2$ | $( 1, 8)( 2, 9)( 3,10)( 6,13)$ | |
$ 2, 2, 2, 2, 2, 2, 1, 1 $ | $7$ | $2$ | $( 1, 8)( 2, 9)( 3,10)( 5,12)( 6,13)( 7,14)$ | |
$ 7, 7 $ | $384$ | $7$ | $( 1,13, 4, 9, 7,12,10)( 2,14, 5, 3, 8, 6,11)$ | |
$ 3, 3, 3, 3, 1, 1 $ | $112$ | $3$ | $( 2,10, 5)( 3,12, 9)( 4,14, 6)( 7,13,11)$ | |
$ 6, 3, 3, 2 $ | $224$ | $6$ | $( 1, 8)( 2,10, 5, 9, 3,12)( 4,14, 6)( 7,13,11)$ | |
$ 6, 6, 1, 1 $ | $112$ | $6$ | $( 2,10,12, 9, 3, 5)( 4,14,13,11, 7, 6)$ | |
$ 3, 3, 3, 3, 1, 1 $ | $112$ | $3$ | $( 2, 5,10)( 3, 9,12)( 4, 6,14)( 7,11,13)$ | |
$ 6, 3, 3, 2 $ | $224$ | $6$ | $( 1, 8)( 2, 5,10, 9,12, 3)( 4, 6,14)( 7,11,13)$ | |
$ 6, 6, 1, 1 $ | $112$ | $6$ | $( 2,12, 3, 9, 5,10)( 4,13, 7,11, 6,14)$ | |
$ 4, 2, 2, 2, 2, 1, 1 $ | $168$ | $4$ | $( 2, 7)( 3, 6)( 4,12,11, 5)( 9,14)(10,13)$ | |
$ 4, 4, 2, 2, 2 $ | $168$ | $4$ | $( 1, 8)( 2, 7, 9,14)( 3, 6)( 4,12,11, 5)(10,13)$ | |
$ 4, 4, 4, 1, 1 $ | $56$ | $4$ | $( 2,14, 9, 7)( 3,13,10, 6)( 4,12,11, 5)$ | |
$ 2, 2, 2, 2, 2, 2, 2 $ | $56$ | $2$ | $( 1, 8)( 2,14)( 3, 6)( 4, 5)( 7, 9)(10,13)(11,12)$ | |
$ 12, 1, 1 $ | $224$ | $12$ | $( 2,13, 5,14, 3, 4, 9, 6,12, 7,10,11)$ | |
$ 6, 6, 2 $ | $224$ | $6$ | $( 1, 8)( 2,13, 5,14, 3, 4)( 6,12, 7,10,11, 9)$ | |
$ 12, 1, 1 $ | $224$ | $12$ | $( 2,11,10, 7, 5, 6, 9, 4, 3,14,12,13)$ | |
$ 6, 6, 2 $ | $224$ | $6$ | $( 1, 8)( 2,11,10, 7, 5, 6)( 3,14,12,13, 9, 4)$ |
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: | yes | magma: IsSolvable(G);
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Nilpotency class: | not nilpotent | ||
Label: | 2688.cb | magma: IdentifyGroup(G);
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Character table: |
Size | |
2 P | |
3 P | |
7 P | |
Type |
magma: CharacterTable(G);