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Magma
magma: G := TransitiveGroup(8, 15);
Group action invariants
Degree $n$: | $8$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $15$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $Z_8 : Z_8^\times$ | ||
CHM label: | $[1/4.cD(4)^{2}]2$ | ||
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: | (1,2,3,4,5,6,7,8), (1,5)(3,7), (1,6)(2,5)(3,4)(7,8) | 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 $8$: $D_{4}$ x 2, $C_2^3$ $16$: $D_4\times C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 4: $D_{4}$
Low degree siblings
8T15, 16T35, 16T38 x 2, 16T45, 32T21Siblings are shown with degree $\leq 47$
A number field with this Galois group has exactly one arithmetically equivalent field.
Conjugacy classes
Label | Cycle Type | Size | Order | Representative |
$ 1, 1, 1, 1, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ | |
$ 2, 2, 2, 1, 1 $ | $4$ | $2$ | $(2,4)(3,7)(6,8)$ | |
$ 2, 2, 1, 1, 1, 1 $ | $2$ | $2$ | $(2,6)(4,8)$ | |
$ 2, 2, 2, 1, 1 $ | $4$ | $2$ | $(2,8)(3,7)(4,6)$ | |
$ 2, 2, 2, 2 $ | $4$ | $2$ | $(1,2)(3,8)(4,7)(5,6)$ | |
$ 8 $ | $4$ | $8$ | $(1,2,3,4,5,6,7,8)$ | |
$ 4, 4 $ | $4$ | $4$ | $(1,2,5,6)(3,8,7,4)$ | |
$ 8 $ | $4$ | $8$ | $(1,2,7,8,5,6,3,4)$ | |
$ 4, 4 $ | $2$ | $4$ | $(1,3,5,7)(2,4,6,8)$ | |
$ 4, 4 $ | $2$ | $4$ | $(1,3,5,7)(2,8,6,4)$ | |
$ 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: | 32.43 | magma: IdentifyGroup(G);
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Character table: |
1A | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 8A | 8B | ||
Size | 1 | 1 | 2 | 4 | 4 | 4 | 2 | 2 | 4 | 4 | 4 | |
2 P | 1A | 1A | 1A | 1A | 1A | 1A | 2A | 2A | 2A | 4A | 4A | |
Type | ||||||||||||
32.43.1a | R | |||||||||||
32.43.1b | R | |||||||||||
32.43.1c | R | |||||||||||
32.43.1d | R | |||||||||||
32.43.1e | R | |||||||||||
32.43.1f | R | |||||||||||
32.43.1g | R | |||||||||||
32.43.1h | R | |||||||||||
32.43.2a | R | |||||||||||
32.43.2b | R | |||||||||||
32.43.4a | R |
magma: CharacterTable(G);