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Magma
magma: G := TransitiveGroup(8, 44);
Group action invariants
Degree $n$: | $8$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $44$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_2 \wr S_4$ | ||
CHM label: | $[2^{4}]S(4)$ | ||
Parity: | $-1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $2$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,2,3,8)(4,5,6,7), (1,8)(4,5), (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 $4$: $C_2^2$ $6$: $S_3$ $12$: $D_{6}$ $24$: $S_4$ $48$: $S_4\times C_2$ $192$: $V_4^2:(S_3\times C_2)$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 4: $S_4$
Low degree siblings
8T44 x 3, 16T736 x 2, 16T743 x 2, 16T748 x 2, 16T752 x 2, 24T708 x 4, 24T1151 x 4, 32T9340, 32T9355, 32T9459 x 4Siblings 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, 1, 1, 1, 1, 1, 1 $ | $4$ | $2$ | $(4,8)$ |
$ 2, 2, 1, 1, 1, 1 $ | $12$ | $2$ | $(3,4)(7,8)$ |
$ 4, 1, 1, 1, 1 $ | $12$ | $4$ | $(3,4,7,8)$ |
$ 2, 2, 1, 1, 1, 1 $ | $6$ | $2$ | $(3,7)(4,8)$ |
$ 2, 2, 2, 1, 1 $ | $24$ | $2$ | $(2,3)(4,8)(6,7)$ |
$ 3, 3, 1, 1 $ | $32$ | $3$ | $(2,3,4)(6,7,8)$ |
$ 6, 1, 1 $ | $32$ | $6$ | $(2,3,4,6,7,8)$ |
$ 4, 2, 1, 1 $ | $24$ | $4$ | $(2,3,6,7)(4,8)$ |
$ 2, 2, 2, 1, 1 $ | $4$ | $2$ | $(2,6)(3,7)(4,8)$ |
$ 2, 2, 2, 2 $ | $12$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ |
$ 4, 2, 2 $ | $24$ | $4$ | $(1,2)(3,4,7,8)(5,6)$ |
$ 2, 2, 2, 2 $ | $12$ | $2$ | $(1,2)(3,7)(4,8)(5,6)$ |
$ 3, 3, 2 $ | $32$ | $6$ | $(1,2,3)(4,8)(5,6,7)$ |
$ 4, 4 $ | $48$ | $4$ | $(1,2,3,4)(5,6,7,8)$ |
$ 8 $ | $48$ | $8$ | $(1,2,3,4,5,6,7,8)$ |
$ 6, 2 $ | $32$ | $6$ | $(1,2,3,5,6,7)(4,8)$ |
$ 4, 4 $ | $12$ | $4$ | $(1,2,5,6)(3,4,7,8)$ |
$ 4, 2, 2 $ | $12$ | $4$ | $(1,2,5,6)(3,7)(4,8)$ |
$ 2, 2, 2, 2 $ | $1$ | $2$ | $(1,5)(2,6)(3,7)(4,8)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $384=2^{7} \cdot 3$ | 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: | 384.5602 | magma: IdentifyGroup(G);
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Character table: |
2 7 5 5 5 6 4 2 2 4 5 5 4 5 2 3 3 2 5 5 7 3 1 1 . . . . 1 1 . 1 . . . 1 . . 1 . . 1 1a 2a 2b 4a 2c 2d 3a 6a 4b 2e 2f 4c 2g 6b 4d 8a 6c 4e 4f 2h 2P 1a 1a 1a 2c 1a 1a 3a 3a 2c 1a 1a 2c 1a 3a 2f 4e 3a 2h 2c 1a 3P 1a 2a 2b 4a 2c 2d 1a 2e 4b 2e 2f 4c 2g 2a 4d 8a 2h 4e 4f 2h 5P 1a 2a 2b 4a 2c 2d 3a 6a 4b 2e 2f 4c 2g 6b 4d 8a 6c 4e 4f 2h 7P 1a 2a 2b 4a 2c 2d 3a 6a 4b 2e 2f 4c 2g 6b 4d 8a 6c 4e 4f 2h X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 -1 -1 1 1 1 1 -1 -1 -1 1 -1 -1 -1 -1 1 1 1 1 1 X.3 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 1 -1 1 X.4 1 1 -1 -1 1 -1 1 1 -1 1 1 1 -1 1 -1 -1 1 1 -1 1 X.5 2 -2 . . 2 . -1 1 . -2 2 -2 . 1 . . -1 2 . 2 X.6 2 2 . . 2 . -1 -1 . 2 2 2 . -1 . . -1 2 . 2 X.7 3 -3 -1 1 3 1 . . -1 -3 -1 1 -1 . 1 -1 . -1 1 3 X.8 3 -3 1 -1 3 -1 . . 1 -3 -1 1 1 . -1 1 . -1 -1 3 X.9 3 3 -1 -1 3 -1 . . -1 3 -1 -1 -1 . 1 1 . -1 -1 3 X.10 3 3 1 1 3 1 . . 1 3 -1 -1 1 . -1 -1 . -1 1 3 X.11 4 -2 2 -2 . . 1 -1 . 2 . . -2 1 . . -1 . 2 -4 X.12 4 -2 -2 2 . . 1 -1 . 2 . . 2 1 . . -1 . -2 -4 X.13 4 2 2 2 . . 1 1 . -2 . . -2 -1 . . -1 . -2 -4 X.14 4 2 -2 -2 . . 1 1 . -2 . . 2 -1 . . -1 . 2 -4 X.15 6 . -2 . -2 . . . 2 . 2 . -2 . . . . -2 . 6 X.16 6 . 2 . -2 . . . -2 . 2 . 2 . . . . -2 . 6 X.17 6 . . -2 -2 2 . . . . -2 . . . . . . 2 -2 6 X.18 6 . . 2 -2 -2 . . . . -2 . . . . . . 2 2 6 X.19 8 -4 . . . . -1 1 . 4 . . . -1 . . 1 . . -8 X.20 8 4 . . . . -1 -1 . -4 . . . 1 . . 1 . . -8 |
magma: CharacterTable(G);