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Magma
magma: G := TransitiveGroup(8, 48);
Group action invariants
Degree $n$: | $8$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $48$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_2^3:\GL(3,2)$ | ||
CHM label: | $E(8):L_{7}=AL(8)$ | ||
Parity: | $1$ | magma: IsEven(G);
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Primitive: | yes | 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: | (1,3)(2,8)(4,6)(5,7), (1,2,6,3,4,5,7), (1,2)(5,6), (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) $168$: $\GL(3,2)$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 4: None
Low degree siblings
8T48, 14T34 x 2, 28T153, 28T159 x 2, 42T210 x 2, 42T211 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, 2, 2 $ | $7$ | $2$ | $(1,3)(2,8)(4,6)(5,7)$ |
$ 2, 2, 1, 1, 1, 1 $ | $42$ | $2$ | $(4,5)(6,7)$ |
$ 2, 2, 2, 2 $ | $42$ | $2$ | $(1,3)(2,8)(4,7)(5,6)$ |
$ 4, 4 $ | $84$ | $4$ | $(1,5,8,4)(2,6,3,7)$ |
$ 3, 3, 1, 1 $ | $224$ | $3$ | $(3,4,5)(6,8,7)$ |
$ 6, 2 $ | $224$ | $6$ | $(1,3,6,2,8,5)(4,7)$ |
$ 4, 2, 1, 1 $ | $168$ | $4$ | $(3,4,8,7)(5,6)$ |
$ 4, 4 $ | $168$ | $4$ | $(1,3,6,7)(2,8,5,4)$ |
$ 7, 1 $ | $192$ | $7$ | $(2,3,4,7,5,8,6)$ |
$ 7, 1 $ | $192$ | $7$ | $(2,3,4,8,6,5,7)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $1344=2^{6} \cdot 3 \cdot 7$ | 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: | no | magma: IsSolvable(G);
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Nilpotency class: | not nilpotent | ||
Label: | 1344.11686 | magma: IdentifyGroup(G);
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Character table: |
2 6 6 5 5 4 1 1 3 3 . . 3 1 1 . . . 1 1 . . . . 7 1 . . . . . . . . 1 1 1a 2a 2b 2c 4a 3a 6a 4b 4c 7a 7b 2P 1a 1a 1a 1a 2a 3a 3a 2b 2c 7a 7b 3P 1a 2a 2b 2c 4a 1a 2a 4b 4c 7b 7a 5P 1a 2a 2b 2c 4a 3a 6a 4b 4c 7b 7a 7P 1a 2a 2b 2c 4a 3a 6a 4b 4c 1a 1a X.1 1 1 1 1 1 1 1 1 1 1 1 X.2 3 3 -1 -1 -1 . . 1 1 A /A X.3 3 3 -1 -1 -1 . . 1 1 /A A X.4 6 6 2 2 2 . . . . -1 -1 X.5 7 -1 3 -1 -1 1 -1 1 -1 . . X.6 7 7 -1 -1 -1 1 1 -1 -1 . . X.7 7 -1 -1 3 -1 1 -1 -1 1 . . X.8 8 8 . . . -1 -1 . . 1 1 X.9 14 -2 2 2 -2 -1 1 . . . . X.10 21 -3 1 -3 1 . . -1 1 . . X.11 21 -3 -3 1 1 . . 1 -1 . . A = E(7)^3+E(7)^5+E(7)^6 = (-1-Sqrt(-7))/2 = -1-b7 |
magma: CharacterTable(G);