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Magma
magma: G := TransitiveGroup(14, 44);
Group action invariants
Degree $n$: | $14$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $44$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_2\wr C_7:C_3$ | ||
CHM label: | $[2^{7}]F_{21}(7)=2wrF_{21}(7)$ | ||
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), (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) $2$: $C_2$ $3$: $C_3$ $6$: $C_6$ $21$: $C_7:C_3$ $42$: $(C_7:C_3) \times C_2$ $168$: $C_2^3:(C_7: C_3)$ x 2 $336$: 14T18 x 2 $1344$: 14T35 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 7: $C_7:C_3$
Low degree siblings
28T226, 28T235 x 2, 28T236, 42T309, 42T310, 42T311Siblings 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $21$ | $2$ | $( 3,10)( 7,14)$ |
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $7$ | $2$ | $( 3,10)( 5,12)( 6,13)( 7,14)$ |
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $21$ | $2$ | $( 2, 9)( 3,10)( 6,13)( 7,14)$ |
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $7$ | $2$ | $( 2, 9)( 5,12)( 6,13)( 7,14)$ |
$ 2, 2, 2, 2, 2, 2, 1, 1 $ | $7$ | $2$ | $( 2, 9)( 3,10)( 4,11)( 5,12)( 6,13)( 7,14)$ |
$ 7, 7 $ | $192$ | $7$ | $( 1,13,11, 9, 7, 5, 3)( 2,14,12,10, 8, 6, 4)$ |
$ 7, 7 $ | $192$ | $7$ | $( 1, 9, 3,11, 5,13, 7)( 2,10, 4,12, 6,14, 8)$ |
$ 3, 3, 3, 3, 1, 1 $ | $112$ | $3$ | $( 1,13, 9)( 2, 8, 6)( 4,12,14)( 5, 7,11)$ |
$ 6, 3, 3, 2 $ | $112$ | $6$ | $( 1,13, 9)( 2, 8, 6)( 3,10)( 4,12, 7,11, 5,14)$ |
$ 6, 6, 1, 1 $ | $112$ | $6$ | $( 1, 6, 2, 8,13, 9)( 4,12, 7,11, 5,14)$ |
$ 6, 3, 3, 2 $ | $112$ | $6$ | $( 1, 6, 2, 8,13, 9)( 3,10)( 4,12,14)( 5, 7,11)$ |
$ 3, 3, 3, 3, 1, 1 $ | $112$ | $3$ | $( 1, 9,13)( 2, 6, 8)( 4,14,12)( 5,11, 7)$ |
$ 6, 3, 3, 2 $ | $112$ | $6$ | $( 1, 9,13)( 2, 6, 8)( 3,10)( 4, 7, 5,11,14,12)$ |
$ 6, 6, 1, 1 $ | $112$ | $6$ | $( 1, 9, 6, 8, 2,13)( 4, 7, 5,11,14,12)$ |
$ 6, 3, 3, 2 $ | $112$ | $6$ | $( 1, 9, 6, 8, 2,13)( 3,10)( 4,14,12)( 5,11, 7)$ |
$ 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $7$ | $2$ | $( 7,14)$ |
$ 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $21$ | $2$ | $( 3,10)( 6,13)( 7,14)$ |
$ 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $7$ | $2$ | $( 3,10)( 5,12)( 6,13)$ |
$ 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $7$ | $2$ | $( 2, 9)( 6,13)( 7,14)$ |
$ 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $21$ | $2$ | $( 2, 9)( 3,10)( 5,12)( 6,13)( 7,14)$ |
$ 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 8)( 2, 9)( 3,10)( 4,11)( 5,12)( 6,13)( 7,14)$ |
$ 14 $ | $192$ | $14$ | $( 1,13,11, 9, 7,12,10, 8, 6, 4, 2,14, 5, 3)$ |
$ 14 $ | $192$ | $14$ | $( 1, 9, 3,11, 5,13, 7, 8, 2,10, 4,12, 6,14)$ |
$ 6, 3, 3, 1, 1 $ | $112$ | $6$ | $( 1,13, 9)( 2, 8, 6)( 4,12,14,11, 5, 7)$ |
$ 3, 3, 3, 3, 2 $ | $112$ | $6$ | $( 1,13, 9)( 2, 8, 6)( 3,10)( 4,12, 7)( 5,14,11)$ |
$ 6, 3, 3, 1, 1 $ | $112$ | $6$ | $( 1, 6, 2, 8,13, 9)( 4,12, 7)( 5,14,11)$ |
$ 6, 6, 2 $ | $112$ | $6$ | $( 1, 6, 2, 8,13, 9)( 3,10)( 4,12,14,11, 5, 7)$ |
$ 6, 3, 3, 1, 1 $ | $112$ | $6$ | $( 1, 9,13)( 2, 6, 8)( 4,14, 5,11, 7,12)$ |
$ 3, 3, 3, 3, 2 $ | $112$ | $6$ | $( 1, 9,13)( 2, 6, 8)( 3,10)( 4, 7,12)( 5,11,14)$ |
$ 6, 3, 3, 1, 1 $ | $112$ | $6$ | $( 1, 9, 6, 8, 2,13)( 4, 7,12)( 5,11,14)$ |
$ 6, 6, 2 $ | $112$ | $6$ | $( 1, 9, 6, 8, 2,13)( 3,10)( 4,14, 5,11, 7,12)$ |
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.ce | magma: IdentifyGroup(G);
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Character table: not available. |
magma: CharacterTable(G);