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Magma
magma: G := TransitiveGroup(24, 46);
Group action invariants
Degree $n$: | $24$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $46$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_2\times S_4$ | ||
Parity: | $1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $8$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (5,17)(6,18)(7,8)(9,10)(11,23)(12,24)(19,20)(21,22), (1,3)(2,4)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(13,15)(14,16), (1,9,18)(2,10,17)(3,11,20)(4,12,19)(5,14,22)(6,13,21)(7,16,24)(8,15,23) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_2^2$ $6$: $S_3$ $12$: $D_{6}$ $24$: $S_4$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: $S_3$
Degree 4: None
Degree 6: $S_3$, $S_4$, $S_4$, $S_4\times C_2$ x 2
Degree 8: None
Degree 12: $S_4$, $C_2\times S_4$, $C_2 \times S_4$
Low degree siblings
6T11 x 2, 8T24 x 2, 12T21, 12T22, 12T23 x 2, 12T24 x 2, 16T61, 24T47, 24T48 x 2Siblings 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 $ | $1$ | $1$ | $()$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $3$ | $2$ | $( 5,17)( 6,18)( 7, 8)( 9,10)(11,23)(12,24)(19,20)(21,22)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $3$ | $2$ | $( 1, 2)( 3,15)( 4,16)( 5,17)( 6,18)( 7, 8)( 9,22)(10,21)(11,24)(12,23)(13,14) (19,20)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $6$ | $2$ | $( 1, 3)( 2, 4)( 5,12)( 6,11)( 7,21)( 8,22)( 9,19)(10,20)(13,15)(14,16)(17,24) (18,23)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $6$ | $2$ | $( 1, 3)( 2, 4)( 5,24)( 6,23)( 7,22)( 8,21)( 9,20)(10,19)(11,18)(12,17)(13,15) (14,16)$ |
$ 4, 4, 4, 4, 4, 4 $ | $6$ | $4$ | $( 1, 4,14,15)( 2, 3,13,16)( 5,12, 6,11)( 7,21,19, 9)( 8,22,20,10)(17,24,18,23)$ |
$ 4, 4, 4, 4, 4, 4 $ | $6$ | $4$ | $( 1, 4,14,15)( 2, 3,13,16)( 5,24, 6,23)( 7,22,19,10)( 8,21,20, 9)(11,17,12,18)$ |
$ 6, 6, 6, 6 $ | $8$ | $6$ | $( 1, 5, 9,14,18,22)( 2, 6,10,13,17,21)( 3, 7,11,16,20,24)( 4, 8,12,15,19,23)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3 $ | $8$ | $3$ | $( 1, 5,21)( 2, 6,22)( 3, 7,12)( 4, 8,11)( 9,13,17)(10,14,18)(15,19,24) (16,20,23)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1,14)( 2,13)( 3,16)( 4,15)( 5,18)( 6,17)( 7,20)( 8,19)( 9,22)(10,21)(11,24) (12,23)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $48=2^{4} \cdot 3$ | 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: | 48.48 | magma: IdentifyGroup(G);
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Character table: |
2 4 4 4 3 3 3 3 1 1 4 3 1 . . . . . . 1 1 1 1a 2a 2b 2c 2d 4a 4b 6a 3a 2e 2P 1a 1a 1a 1a 1a 2b 2b 3a 3a 1a 3P 1a 2a 2b 2c 2d 4a 4b 2e 1a 2e 5P 1a 2a 2b 2c 2d 4a 4b 6a 3a 2e X.1 1 1 1 1 1 1 1 1 1 1 X.2 1 -1 1 -1 1 1 -1 -1 1 -1 X.3 1 -1 1 1 -1 -1 1 -1 1 -1 X.4 1 1 1 -1 -1 -1 -1 1 1 1 X.5 2 -2 2 . . . . 1 -1 -2 X.6 2 2 2 . . . . -1 -1 2 X.7 3 -1 -1 -1 -1 1 1 . . 3 X.8 3 -1 -1 1 1 -1 -1 . . 3 X.9 3 1 -1 -1 1 -1 1 . . -3 X.10 3 1 -1 1 -1 1 -1 . . -3 |
magma: CharacterTable(G);