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Magma
magma: G := TransitiveGroup(28, 42);
Group action invariants
Degree $n$: | $28$ | 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: | $\PGL(2,7)$ | ||
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: | (1,14,21,9,15,6,4)(2,13,22,10,16,5,3)(7,11,26,17,20,28,23)(8,12,25,18,19,27,24), (1,18,13,28,22,11,5,26)(2,17,14,27,21,12,6,25)(3,19,4,20)(7,9,23,16,8,10,24,15) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 4: None
Degree 7: None
Degree 14: 14T16
Low degree siblings
8T43, 14T16, 16T713, 21T20, 24T707, 28T46, 42T81, 42T82, 42T83Siblings 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 $ | $1$ | $1$ | $()$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $21$ | $2$ | $( 3, 4)( 5,15)( 6,16)( 9,14)(10,13)(11,12)(17,25)(18,26)(19,24)(20,23)(21,22) (27,28)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1 $ | $56$ | $3$ | $( 3, 5,10)( 4, 6, 9)( 7,28,11)( 8,27,12)(13,22,16)(14,21,15)(17,20,23) (18,19,24)$ |
$ 4, 4, 4, 4, 4, 4, 2, 2 $ | $42$ | $4$ | $( 1, 2)( 3, 5,21,16)( 4, 6,22,15)( 7,28, 8,27)( 9,14,10,13)(11,12) (17,26,20,24)(18,25,19,23)$ |
$ 7, 7, 7, 7 $ | $48$ | $7$ | $( 1, 3, 6,22,13,10,15)( 2, 4, 5,21,14, 9,16)( 7,23,18,26,12,19,28) ( 8,24,17,25,11,20,27)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $28$ | $2$ | $( 1, 7)( 2, 8)( 3,11)( 4,12)( 5,17)( 6,18)( 9,23)(10,24)(13,19)(14,20)(15,25) (16,26)(21,27)(22,28)$ |
$ 8, 8, 8, 4 $ | $42$ | $8$ | $( 1, 7,22,26,16,19,10,11)( 2, 8,21,25,15,20, 9,12)( 3,17,14,27, 4,18,13,28) ( 5,24, 6,23)$ |
$ 6, 6, 6, 6, 2, 2 $ | $56$ | $6$ | $( 1, 7,22,25,14,12)( 2, 8,21,26,13,11)( 3,18,16,23, 9,27)( 4,17,15,24,10,28) ( 5,20)( 6,19)$ |
$ 8, 8, 8, 4 $ | $42$ | $8$ | $( 1, 7, 2, 8)( 3,27,22,12, 4,28,21,11)( 5,26,10,23,15,18,13,20) ( 6,25, 9,24,16,17,14,19)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $336=2^{4} \cdot 3 \cdot 7$ | 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: | 336.208 | magma: IdentifyGroup(G);
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Character table: |
2 4 4 1 3 . 2 3 1 3 3 1 . 1 . . 1 . 1 . 7 1 . . . 1 . . . . 1a 2a 3a 4a 7a 2b 8a 6a 8b 2P 1a 1a 3a 2a 7a 1a 4a 3a 4a 3P 1a 2a 1a 4a 7a 2b 8b 2b 8a 5P 1a 2a 3a 4a 7a 2b 8b 6a 8a 7P 1a 2a 3a 4a 1a 2b 8a 6a 8b X.1 1 1 1 1 1 1 1 1 1 X.2 1 1 1 1 1 -1 -1 -1 -1 X.3 6 -2 . 2 -1 . . . . X.4 6 2 . . -1 . A . -A X.5 6 2 . . -1 . -A . A X.6 7 -1 1 -1 . -1 1 -1 1 X.7 7 -1 1 -1 . 1 -1 1 -1 X.8 8 . -1 . 1 -2 . 1 . X.9 8 . -1 . 1 2 . -1 . A = -E(8)+E(8)^3 = -Sqrt(2) = -r2 |
magma: CharacterTable(G);