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Magma
magma: G := TransitiveGroup(8, 3);
Group action invariants
Degree $n$: | $8$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $3$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_2^3$ | ||
CHM label: | $E(8)=2[x]2[x]2$ | ||
Parity: | $1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $8$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,3)(2,8)(4,6)(5,7), (1,8)(2,3)(4,5)(6,7), (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 7 $4$: $C_2^2$ x 7 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$ x 7
Degree 4: $C_2^2$ x 7
Low degree siblings
There are no siblings 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 $ | $1$ | $2$ | $(1,2)(3,8)(4,7)(5,6)$ |
$ 2, 2, 2, 2 $ | $1$ | $2$ | $(1,3)(2,8)(4,6)(5,7)$ |
$ 2, 2, 2, 2 $ | $1$ | $2$ | $(1,4)(2,7)(3,6)(5,8)$ |
$ 2, 2, 2, 2 $ | $1$ | $2$ | $(1,5)(2,6)(3,7)(4,8)$ |
$ 2, 2, 2, 2 $ | $1$ | $2$ | $(1,6)(2,5)(3,4)(7,8)$ |
$ 2, 2, 2, 2 $ | $1$ | $2$ | $(1,7)(2,4)(3,5)(6,8)$ |
$ 2, 2, 2, 2 $ | $1$ | $2$ | $(1,8)(2,3)(4,5)(6,7)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $8=2^{3}$ | magma: Order(G);
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Cyclic: | no | magma: IsCyclic(G);
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Abelian: | yes | magma: IsAbelian(G);
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Solvable: | yes | magma: IsSolvable(G);
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Nilpotency class: | $1$ | ||
Label: | 8.5 | magma: IdentifyGroup(G);
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Character table: |
2 3 3 3 3 3 3 3 3 1a 2a 2b 2c 2d 2e 2f 2g 2P 1a 1a 1a 1a 1a 1a 1a 1a X.1 1 1 1 1 1 1 1 1 X.2 1 -1 -1 -1 -1 1 1 1 X.3 1 -1 -1 1 1 -1 -1 1 X.4 1 -1 1 -1 1 -1 1 -1 X.5 1 -1 1 1 -1 1 -1 -1 X.6 1 1 -1 -1 1 1 -1 -1 X.7 1 1 -1 1 -1 -1 1 -1 X.8 1 1 1 -1 -1 -1 -1 1 |
magma: CharacterTable(G);