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Magma
magma: G := TransitiveGroup(14, 51);
Group action invariants
Degree $n$: | $14$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $51$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_2^7.\GL(3,2)$ | ||
CHM label: | $[2^{7}]L(7)=2wrL(7)$ | ||
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), (2,4)(5,13)(6,12)(9,11), (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) $2$: $C_2$ $168$: $\GL(3,2)$ $336$: 14T17 $1344$: $C_2^3:\GL(3,2)$ $2688$: 14T43 $10752$: 14T50 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 7: $\GL(3,2)$
Low degree siblings
14T51, 28T462 x 2, 28T463, 28T464 x 2, 42T767 x 2, 42T768 x 2, 42T769 x 2, 42T770 x 2Siblings 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $7$ | $2$ | $( 7,14)$ | |
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $21$ | $2$ | $( 2, 9)( 7,14)$ | |
$ 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $28$ | $2$ | $( 2, 9)( 4,11)( 7,14)$ | |
$ 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $7$ | $2$ | $( 2, 9)( 6,13)( 7,14)$ | |
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $28$ | $2$ | $( 2, 9)( 4,11)( 6,13)( 7,14)$ | |
$ 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, 1, 1, 1, 1 $ | $21$ | $2$ | $( 2, 9)( 4,11)( 5,12)( 6,13)( 7,14)$ | |
$ 2, 2, 2, 2, 2, 2, 1, 1 $ | $7$ | $2$ | $( 2, 9)( 3,10)( 4,11)( 5,12)( 6,13)( 7,14)$ | |
$ 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 8)( 2, 9)( 3,10)( 4,11)( 5,12)( 6,13)( 7,14)$ | |
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $84$ | $2$ | $( 3, 5)( 6, 7)(10,12)(13,14)$ | |
$ 4, 2, 2, 1, 1, 1, 1, 1, 1 $ | $168$ | $4$ | $( 3, 5)( 6,14,13, 7)(10,12)$ | |
$ 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $84$ | $2$ | $( 2, 9)( 3, 5)( 6, 7)(10,12)(13,14)$ | |
$ 4, 2, 2, 2, 1, 1, 1, 1 $ | $168$ | $4$ | $( 2, 9)( 3, 5)( 6,14,13, 7)(10,12)$ | |
$ 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $168$ | $2$ | $( 3, 5)( 4,11)( 6, 7)(10,12)(13,14)$ | |
$ 4, 2, 2, 2, 1, 1, 1, 1 $ | $336$ | $4$ | $( 3, 5)( 4,11)( 6,14,13, 7)(10,12)$ | |
$ 2, 2, 2, 2, 2, 2, 1, 1 $ | $168$ | $2$ | $( 2, 9)( 3, 5)( 4,11)( 6, 7)(10,12)(13,14)$ | |
$ 4, 2, 2, 2, 2, 1, 1 $ | $336$ | $4$ | $( 2, 9)( 3, 5)( 4,11)( 6,14,13, 7)(10,12)$ | |
$ 4, 4, 1, 1, 1, 1, 1, 1 $ | $84$ | $4$ | $( 3,12,10, 5)( 6,14,13, 7)$ | |
$ 4, 4, 2, 1, 1, 1, 1 $ | $84$ | $4$ | $( 2, 9)( 3,12,10, 5)( 6,14,13, 7)$ | |
$ 4, 4, 2, 1, 1, 1, 1 $ | $168$ | $4$ | $( 3,12,10, 5)( 4,11)( 6,14,13, 7)$ | |
$ 4, 4, 2, 2, 1, 1 $ | $168$ | $4$ | $( 2, 9)( 3,12,10, 5)( 4,11)( 6,14,13, 7)$ | |
$ 2, 2, 2, 2, 2, 2, 1, 1 $ | $84$ | $2$ | $( 1, 8)( 3, 5)( 4,11)( 6, 7)(10,12)(13,14)$ | |
$ 4, 2, 2, 2, 2, 1, 1 $ | $168$ | $4$ | $( 1, 8)( 3, 5)( 4,11)( 6,14,13, 7)(10,12)$ | |
$ 2, 2, 2, 2, 2, 2, 2 $ | $84$ | $2$ | $( 1, 8)( 2, 9)( 3, 5)( 4,11)( 6, 7)(10,12)(13,14)$ | |
$ 4, 2, 2, 2, 2, 2 $ | $168$ | $4$ | $( 1, 8)( 2, 9)( 3, 5)( 4,11)( 6,14,13, 7)(10,12)$ | |
$ 4, 4, 2, 2, 1, 1 $ | $84$ | $4$ | $( 1, 8)( 3,12,10, 5)( 4,11)( 6,14,13, 7)$ | |
$ 4, 4, 2, 2, 2 $ | $84$ | $4$ | $( 1, 8)( 2, 9)( 3,12,10, 5)( 4,11)( 6,14,13, 7)$ | |
$ 4, 4, 2, 2, 1, 1 $ | $672$ | $4$ | $( 2, 3, 4, 7)( 5, 6)( 9,10,11,14)(12,13)$ | |
$ 8, 2, 2, 1, 1 $ | $672$ | $8$ | $( 2, 3, 4,14, 9,10,11, 7)( 5, 6)(12,13)$ | |
$ 4, 4, 4, 1, 1 $ | $672$ | $4$ | $( 2, 3, 4, 7)( 5,13,12, 6)( 9,10,11,14)$ | |
$ 8, 4, 1, 1 $ | $672$ | $8$ | $( 2, 3, 4,14, 9,10,11, 7)( 5,13,12, 6)$ | |
$ 4, 4, 2, 2, 2 $ | $672$ | $4$ | $( 1, 8)( 2, 3, 4, 7)( 5, 6)( 9,10,11,14)(12,13)$ | |
$ 8, 2, 2, 2 $ | $672$ | $8$ | $( 1, 8)( 2, 3, 4,14, 9,10,11, 7)( 5, 6)(12,13)$ | |
$ 4, 4, 4, 2 $ | $672$ | $4$ | $( 1, 8)( 2, 3, 4, 7)( 5,13,12, 6)( 9,10,11,14)$ | |
$ 8, 4, 2 $ | $672$ | $8$ | $( 1, 8)( 2, 3, 4,14, 9,10,11, 7)( 5,13,12, 6)$ | |
$ 3, 3, 3, 3, 1, 1 $ | $896$ | $3$ | $( 2, 3, 5)( 4, 7, 6)( 9,10,12)(11,14,13)$ | |
$ 6, 3, 3, 1, 1 $ | $896$ | $6$ | $( 2, 3, 5)( 4,14,13,11, 7, 6)( 9,10,12)$ | |
$ 6, 3, 3, 1, 1 $ | $896$ | $6$ | $( 2, 3, 5, 9,10,12)( 4, 7, 6)(11,14,13)$ | |
$ 6, 6, 1, 1 $ | $896$ | $6$ | $( 2, 3, 5, 9,10,12)( 4,14,13,11, 7, 6)$ | |
$ 3, 3, 3, 3, 2 $ | $896$ | $6$ | $( 1, 8)( 2, 3, 5)( 4, 7, 6)( 9,10,12)(11,14,13)$ | |
$ 6, 3, 3, 2 $ | $896$ | $6$ | $( 1, 8)( 2, 3, 5)( 4,14,13,11, 7, 6)( 9,10,12)$ | |
$ 6, 3, 3, 2 $ | $896$ | $6$ | $( 1, 8)( 2, 3, 5, 9,10,12)( 4, 7, 6)(11,14,13)$ | |
$ 6, 6, 2 $ | $896$ | $6$ | $( 1, 8)( 2, 3, 5, 9,10,12)( 4,14,13,11, 7, 6)$ | |
$ 7, 7 $ | $1536$ | $7$ | $( 1, 2, 3, 4, 5, 6, 7)( 8, 9,10,11,12,13,14)$ | |
$ 14 $ | $1536$ | $14$ | $( 1, 2, 3, 4, 5, 6,14, 8, 9,10,11,12,13, 7)$ | |
$ 7, 7 $ | $1536$ | $7$ | $( 1, 2, 3, 7, 6, 4, 5)( 8, 9,10,14,13,11,12)$ | |
$ 14 $ | $1536$ | $14$ | $( 1, 2, 3,14,13,11,12, 8, 9,10, 7, 6, 4, 5)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $21504=2^{10} \cdot 3 \cdot 7$ | 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: | 21504.t | magma: IdentifyGroup(G);
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Character table: | 48 x 48 character table |
magma: CharacterTable(G);