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Magma
magma: G := TransitiveGroup(8, 24);
Group action invariants
Degree $n$: | $8$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $24$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $S_4\times C_2$ | ||
CHM label: | $E(8):D_{6}=S(4)[x]2$ | ||
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: | (1,3)(2,8)(4,6)(5,7), (2,3)(4,5), (1,8)(2,3)(4,5)(6,7), (1,2,3)(4,6,5), (1,5)(2,6)(3,7)(4,8) | 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 4: $S_4$
Low degree siblings
6T11 x 2, 8T24, 12T21, 12T22, 12T23 x 2, 12T24 x 2, 16T61, 24T46, 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 $ | $1$ | $1$ | $()$ |
$ 2, 2, 1, 1, 1, 1 $ | $6$ | $2$ | $(3,8)(4,7)$ |
$ 3, 3, 1, 1 $ | $8$ | $3$ | $(2,3,8)(4,7,5)$ |
$ 2, 2, 2, 2 $ | $3$ | $2$ | $(1,2)(3,8)(4,7)(5,6)$ |
$ 4, 4 $ | $6$ | $4$ | $(1,2,3,8)(4,7,6,5)$ |
$ 2, 2, 2, 2 $ | $6$ | $2$ | $(1,4)(2,5)(3,6)(7,8)$ |
$ 6, 2 $ | $8$ | $6$ | $(1,4,8,6,3,7)(2,5)$ |
$ 4, 4 $ | $6$ | $4$ | $(1,4,8,5)(2,6,3,7)$ |
$ 2, 2, 2, 2 $ | $3$ | $2$ | $(1,4)(2,7)(3,6)(5,8)$ |
$ 2, 2, 2, 2 $ | $1$ | $2$ | $(1,6)(2,5)(3,4)(7,8)$ |
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 3 1 4 3 3 1 3 4 4 3 1 . 1 . . . 1 . . 1 1a 2a 3a 2b 4a 2c 6a 4b 2d 2e 2P 1a 1a 3a 1a 2b 1a 3a 2b 1a 1a 3P 1a 2a 1a 2b 4a 2c 2e 4b 2d 2e 5P 1a 2a 3a 2b 4a 2c 6a 4b 2d 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 . -1 2 . . -1 . 2 2 X.6 2 . -1 2 . . 1 . -2 -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);