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Magma
magma: G := TransitiveGroup(8, 23);
Group action invariants
Degree $n$: | $8$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $23$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $\textrm{GL(2,3)}$ | ||
CHM label: | $2S_{4}(8)=GL(2,3)$ | ||
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: | (1,2,3,4,5,6,7,8), (1,3,8)(4,5,7) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $6$: $S_3$ $24$: $S_4$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 4: $S_4$
Low degree siblings
8T23, 16T66, 24T22Siblings are shown with degree $\leq 47$
A number field with this Galois group has exactly one arithmetically equivalent field.
Conjugacy classes
Cycle Type | Size | Order | Representative |
$ 1, 1, 1, 1, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ |
$ 2, 2, 2, 1, 1 $ | $12$ | $2$ | $(2,3)(4,8)(6,7)$ |
$ 3, 3, 1, 1 $ | $8$ | $3$ | $(2,7,8)(3,4,6)$ |
$ 8 $ | $6$ | $8$ | $(1,2,3,4,5,6,7,8)$ |
$ 4, 4 $ | $6$ | $4$ | $(1,2,5,6)(3,8,7,4)$ |
$ 6, 2 $ | $8$ | $6$ | $(1,2,7,5,6,3)(4,8)$ |
$ 8 $ | $6$ | $8$ | $(1,2,8,3,5,6,4,7)$ |
$ 2, 2, 2, 2 $ | $1$ | $2$ | $(1,5)(2,6)(3,7)(4,8)$ |
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: | 48.29 | magma: IdentifyGroup(G);
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Character table: |
2 4 2 1 3 3 1 3 4 3 1 . 1 . . 1 . 1 1a 2a 3a 8a 4a 6a 8b 2b 2P 1a 1a 3a 4a 2b 3a 4a 1a 3P 1a 2a 1a 8a 4a 2b 8b 2b 5P 1a 2a 3a 8b 4a 6a 8a 2b 7P 1a 2a 3a 8b 4a 6a 8a 2b X.1 1 1 1 1 1 1 1 1 X.2 1 -1 1 -1 1 1 -1 1 X.3 2 . -1 . 2 -1 . 2 X.4 2 . -1 A . 1 -A -2 X.5 2 . -1 -A . 1 A -2 X.6 3 1 . -1 -1 . -1 3 X.7 3 -1 . 1 -1 . 1 3 X.8 4 . 1 . . -1 . -4 A = E(8)+E(8)^3 = Sqrt(-2) = i2 |
magma: CharacterTable(G);