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Magma
magma: G := TransitiveGroup(28, 26);
Group action invariants
Degree $n$: | $28$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $26$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_4\times F_7$ | ||
Parity: | $-1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $4$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,9,21,26,18,5)(2,10,22,25,17,6)(3,19,16,24,7,11)(4,20,15,23,8,12), (1,16,2,15)(3,6,23,18,19,10,4,5,24,17,20,9)(7,14,12,21,27,25,8,13,11,22,28,26) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $3$: $C_3$ $4$: $C_4$ x 2, $C_2^2$ $6$: $C_6$ x 3 $8$: $C_4\times C_2$ $12$: $C_{12}$ x 2, $C_6\times C_2$ $24$: 24T2 $42$: $F_7$ $84$: $F_7 \times C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 4: $C_4$
Degree 7: $F_7$
Degree 14: $F_7 \times C_2$
Low degree siblings
28T26Siblings 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ |
$ 6, 6, 6, 6, 1, 1, 1, 1 $ | $7$ | $6$ | $( 3, 7,19,27,24,11)( 4, 8,20,28,23,12)( 5,13, 9,26,18,21)( 6,14,10,25,17,22)$ |
$ 6, 6, 6, 6, 1, 1, 1, 1 $ | $7$ | $6$ | $( 3,11,24,27,19, 7)( 4,12,23,28,20, 8)( 5,21,18,26, 9,13)( 6,22,17,25,10,14)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1 $ | $7$ | $3$ | $( 3,19,24)( 4,20,23)( 5, 9,18)( 6,10,17)( 7,27,11)( 8,28,12)(13,26,21) (14,25,22)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1 $ | $7$ | $3$ | $( 3,24,19)( 4,23,20)( 5,18, 9)( 6,17,10)( 7,11,27)( 8,12,28)(13,21,26) (14,22,25)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $7$ | $2$ | $( 3,27)( 4,28)( 5,26)( 6,25)( 7,24)( 8,23)( 9,21)(10,22)(11,19)(12,20)(13,18) (14,17)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)$ |
$ 6, 6, 6, 6, 2, 2 $ | $7$ | $6$ | $( 1, 2)( 3, 8,19,28,24,12)( 4, 7,20,27,23,11)( 5,14, 9,25,18,22) ( 6,13,10,26,17,21)(15,16)$ |
$ 6, 6, 6, 6, 2, 2 $ | $7$ | $6$ | $( 1, 2)( 3,12,24,28,19, 8)( 4,11,23,27,20, 7)( 5,22,18,25, 9,14) ( 6,21,17,26,10,13)(15,16)$ |
$ 6, 6, 6, 6, 2, 2 $ | $7$ | $6$ | $( 1, 2)( 3,20,24, 4,19,23)( 5,10,18, 6, 9,17)( 7,28,11, 8,27,12) (13,25,21,14,26,22)(15,16)$ |
$ 6, 6, 6, 6, 2, 2 $ | $7$ | $6$ | $( 1, 2)( 3,23,19, 4,24,20)( 5,17, 9, 6,18,10)( 7,12,27, 8,11,28) (13,22,26,14,21,25)(15,16)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $7$ | $2$ | $( 1, 2)( 3,28)( 4,27)( 5,25)( 6,26)( 7,23)( 8,24)( 9,22)(10,21)(11,20)(12,19) (13,17)(14,18)(15,16)$ |
$ 4, 4, 4, 4, 4, 4, 4 $ | $7$ | $4$ | $( 1, 3, 2, 4)( 5,27, 6,28)( 7,25, 8,26)( 9,24,10,23)(11,22,12,21)(13,19,14,20) (15,18,16,17)$ |
$ 28 $ | $6$ | $28$ | $( 1, 3, 6, 8, 9,11,14,15,18,19,22,23,26,27, 2, 4, 5, 7,10,12,13,16,17,20,21, 24,25,28)$ |
$ 12, 12, 4 $ | $7$ | $12$ | $( 1, 3,10,28,26,19, 2, 4, 9,27,25,20)( 5,16,17,23,13,11, 6,15,18,24,14,12) ( 7,22, 8,21)$ |
$ 12, 12, 4 $ | $7$ | $12$ | $( 1, 3,14, 8, 5,24, 2, 4,13, 7, 6,23)( 9,16,17,28,21,19,10,15,18,27,22,20) (11,25,12,26)$ |
$ 12, 12, 4 $ | $7$ | $12$ | $( 1, 3,22,15,18, 7, 2, 4,21,16,17, 8)( 5,11,10,20,26,24, 6,12, 9,19,25,23) (13,27,14,28)$ |
$ 12, 12, 4 $ | $7$ | $12$ | $( 1, 3,25,15,18,11, 2, 4,26,16,17,12)( 5,19, 6,20)( 7,14,23,21,27,10, 8,13,24, 22,28, 9)$ |
$ 4, 4, 4, 4, 4, 4, 4 $ | $7$ | $4$ | $( 1, 4, 2, 3)( 5,28, 6,27)( 7,26, 8,25)( 9,23,10,24)(11,21,12,22)(13,20,14,19) (15,17,16,18)$ |
$ 28 $ | $6$ | $28$ | $( 1, 4, 6, 7, 9,12,14,16,18,20,22,24,26,28, 2, 3, 5, 8,10,11,13,15,17,19,21, 23,25,27)$ |
$ 12, 12, 4 $ | $7$ | $12$ | $( 1, 4,10,27,26,20, 2, 3, 9,28,25,19)( 5,15,17,24,13,12, 6,16,18,23,14,11) ( 7,21, 8,22)$ |
$ 12, 12, 4 $ | $7$ | $12$ | $( 1, 4,14, 7, 5,23, 2, 3,13, 8, 6,24)( 9,15,17,27,21,20,10,16,18,28,22,19) (11,26,12,25)$ |
$ 12, 12, 4 $ | $7$ | $12$ | $( 1, 4,22,16,18, 8, 2, 3,21,15,17, 7)( 5,12,10,19,26,23, 6,11, 9,20,25,24) (13,28,14,27)$ |
$ 12, 12, 4 $ | $7$ | $12$ | $( 1, 4,25,16,18,12, 2, 3,26,15,17,11)( 5,20, 6,19)( 7,13,23,22,27, 9, 8,14,24, 21,28,10)$ |
$ 7, 7, 7, 7 $ | $6$ | $7$ | $( 1, 5, 9,13,18,21,26)( 2, 6,10,14,17,22,25)( 3, 7,11,16,19,24,27) ( 4, 8,12,15,20,23,28)$ |
$ 14, 14 $ | $6$ | $14$ | $( 1, 6, 9,14,18,22,26, 2, 5,10,13,17,21,25)( 3, 8,11,15,19,23,27, 4, 7,12,16, 20,24,28)$ |
$ 4, 4, 4, 4, 4, 4, 4 $ | $1$ | $4$ | $( 1,15, 2,16)( 3,18, 4,17)( 5,20, 6,19)( 7,21, 8,22)( 9,23,10,24)(11,26,12,25) (13,28,14,27)$ |
$ 4, 4, 4, 4, 4, 4, 4 $ | $1$ | $4$ | $( 1,16, 2,15)( 3,17, 4,18)( 5,19, 6,20)( 7,22, 8,21)( 9,24,10,23)(11,25,12,26) (13,27,14,28)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $168=2^{3} \cdot 3 \cdot 7$ | 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: | 168.8 | magma: IdentifyGroup(G);
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Character table: not available. |
magma: CharacterTable(G);