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Magma
magma: G := TransitiveGroup(8, 45);
Group action invariants
Degree $n$: | $8$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $45$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $(A_4\wr C_2):C_2$ | ||
CHM label: | $[1/2.S(4)^{2}]2$ | ||
Parity: | $1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $1$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,8)(4,5), (1,3)(2,8), (1,2,3), (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 $4$: $C_2^2$ $6$: $S_3$ x 2 $12$: $D_{6}$ x 2 $36$: $S_3^2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 4: None
Low degree siblings
12T161, 12T163, 12T165 x 2, 16T1032, 16T1034, 18T179, 18T180, 18T185 x 2, 24T1490, 24T1492, 24T1493 x 2, 24T1494 x 2, 24T1495 x 2, 24T1503, 24T1504 x 2, 32T34597 x 2, 32T34598, 36T759, 36T760, 36T762, 36T763, 36T774 x 2, 36T775 x 2, 36T960, 36T961, 36T962 x 2, 36T963 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 $ | $6$ | $2$ | $(4,5)(6,7)$ |
$ 2, 2, 2, 2 $ | $9$ | $2$ | $(1,8)(2,3)(4,5)(6,7)$ |
$ 3, 1, 1, 1, 1, 1 $ | $16$ | $3$ | $(2,3,8)$ |
$ 3, 2, 2, 1 $ | $48$ | $6$ | $(2,3,8)(4,5)(6,7)$ |
$ 3, 3, 1, 1 $ | $32$ | $3$ | $(2,3,8)(5,7,6)$ |
$ 3, 3, 1, 1 $ | $32$ | $3$ | $(2,8,3)(5,7,6)$ |
$ 2, 2, 1, 1, 1, 1 $ | $36$ | $2$ | $(3,8)(6,7)$ |
$ 4, 2, 1, 1 $ | $72$ | $4$ | $(1,8,2,3)(6,7)$ |
$ 4, 4 $ | $36$ | $4$ | $(1,8,2,3)(4,6,5,7)$ |
$ 2, 2, 2, 2 $ | $12$ | $2$ | $(1,5)(2,6)(3,7)(4,8)$ |
$ 4, 4 $ | $36$ | $4$ | $(1,4,8,5)(2,7,3,6)$ |
$ 6, 2 $ | $96$ | $6$ | $(1,5)(2,6,3,7,8,4)$ |
$ 6, 2 $ | $96$ | $6$ | $(1,5)(2,7,8,4,3,6)$ |
$ 2, 2, 2, 2 $ | $12$ | $2$ | $(1,5)(2,7)(3,6)(4,8)$ |
$ 4, 4 $ | $36$ | $4$ | $(1,4,8,5)(2,6,3,7)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $576=2^{6} \cdot 3^{2}$ | 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: | 576.8654 | magma: IdentifyGroup(G);
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Character table: |
2 6 5 6 2 2 1 1 4 3 4 4 4 1 1 4 4 3 2 1 . 2 1 2 2 . . . 1 . 1 1 1 . 1a 2a 2b 3a 6a 3b 3c 2c 4a 4b 2d 4c 6b 6c 2e 4d 2P 1a 1a 1a 3a 3a 3b 3c 1a 2a 2b 1a 2b 3b 3c 1a 2b 3P 1a 2a 2b 1a 2a 1a 1a 2c 4a 4b 2d 4c 2d 2e 2e 4d 5P 1a 2a 2b 3a 6a 3b 3c 2c 4a 4b 2d 4c 6b 6c 2e 4d X.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 X.3 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 X.5 2 2 2 -1 -1 -1 2 . . . 2 2 -1 . . . X.6 2 2 2 -1 -1 -1 2 . . . -2 -2 1 . . . X.7 2 2 2 -1 -1 2 -1 . . . . . . -1 2 2 X.8 2 2 2 -1 -1 2 -1 . . . . . . 1 -2 -2 X.9 4 4 4 1 1 -2 -2 . . . . . . . . . X.10 6 2 -2 3 -1 . . -2 . 2 . . . . . . X.11 6 2 -2 3 -1 . . 2 . -2 . . . . . . X.12 9 -3 1 . . . . -1 1 -1 -3 1 . . 3 -1 X.13 9 -3 1 . . . . -1 1 -1 3 -1 . . -3 1 X.14 9 -3 1 . . . . 1 -1 1 -3 1 . . -3 1 X.15 9 -3 1 . . . . 1 -1 1 3 -1 . . 3 -1 X.16 12 4 -4 -3 1 . . . . . . . . . . . |
magma: CharacterTable(G);