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Magma
magma: G := TransitiveGroup(8, 21);
Group action invariants
Degree $n$: | $8$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $21$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_2^3: C_4$ | ||
CHM label: | $1/2[2^{4}]E(4)=[1/4.dD(4)^{2}]2$ | ||
Parity: | $-1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $2$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,4,5,8)(2,3)(6,7), (1,3)(2,8)(4,6)(5,7), (1,5)(3,7) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_4$ x 2, $C_2^2$ $8$: $D_{4}$ x 2, $C_4\times C_2$ $16$: $C_2^2:C_4$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$ x 3
Degree 4: $C_2^2$
Low degree siblings
8T19 x 2, 8T20, 16T33 x 2, 16T52, 16T53, 32T19Siblings 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 $ | $2$ | $2$ | $(3,7)(4,8)$ |
$ 2, 2, 1, 1, 1, 1 $ | $2$ | $2$ | $(2,6)(4,8)$ |
$ 2, 2, 1, 1, 1, 1 $ | $2$ | $2$ | $(2,6)(3,7)$ |
$ 4, 2, 2 $ | $4$ | $4$ | $(1,2)(3,4,7,8)(5,6)$ |
$ 4, 2, 2 $ | $4$ | $4$ | $(1,2)(3,8,7,4)(5,6)$ |
$ 2, 2, 2, 2 $ | $4$ | $2$ | $(1,3)(2,4)(5,7)(6,8)$ |
$ 4, 4 $ | $4$ | $4$ | $(1,3,5,7)(2,4,6,8)$ |
$ 4, 2, 2 $ | $4$ | $4$ | $(1,4,5,8)(2,3)(6,7)$ |
$ 4, 2, 2 $ | $4$ | $4$ | $(1,4)(2,3,6,7)(5,8)$ |
$ 2, 2, 2, 2 $ | $1$ | $2$ | $(1,5)(2,6)(3,7)(4,8)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $32=2^{5}$ | 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: | $3$ | ||
Label: | 32.6 | magma: IdentifyGroup(G);
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Character table: |
2 5 4 4 4 3 3 3 3 3 3 5 1a 2a 2b 2c 4a 4b 2d 4c 4d 4e 2e 2P 1a 1a 1a 1a 2a 2a 1a 2e 2c 2c 1a 3P 1a 2a 2b 2c 4b 4a 2d 4c 4e 4d 2e X.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 X.3 1 1 1 1 -1 -1 1 1 -1 -1 1 X.4 1 1 1 1 1 1 -1 -1 -1 -1 1 X.5 1 -1 1 -1 A -A -1 1 -A A 1 X.6 1 -1 1 -1 -A A -1 1 A -A 1 X.7 1 -1 1 -1 A -A 1 -1 A -A 1 X.8 1 -1 1 -1 -A A 1 -1 -A A 1 X.9 2 2 -2 -2 . . . . . . 2 X.10 2 -2 -2 2 . . . . . . 2 X.11 4 . . . . . . . . . -4 A = -E(4) = -Sqrt(-1) = -i |
magma: CharacterTable(G);