Show commands:
Magma
magma: G := TransitiveGroup(8, 22);
Group action invariants
Degree $n$: | $8$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $22$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $Q_8:C_2^2$ | ||
CHM label: | $E(8):D_{4}=[2^{3}]2^{2}$ | ||
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: | (1,3)(2,8)(4,6)(5,7), (2,3)(6,7), (2,3)(4,5), (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 15 $4$: $C_2^2$ x 35 $8$: $C_2^3$ x 15 $16$: $C_2^4$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$ x 3
Degree 4: $C_2^2$
Low degree siblings
8T22 x 5, 16T23 x 9, 32T9Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
Label | Cycle Type | Size | Order | Representative |
$ 1, 1, 1, 1, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ | |
$ 2, 2, 1, 1, 1, 1 $ | $2$ | $2$ | $(4,5)(6,7)$ | |
$ 2, 2, 1, 1, 1, 1 $ | $2$ | $2$ | $(2,3)(6,7)$ | |
$ 2, 2, 1, 1, 1, 1 $ | $2$ | $2$ | $(2,3)(4,5)$ | |
$ 2, 2, 2, 2 $ | $2$ | $2$ | $(1,2)(3,8)(4,6)(5,7)$ | |
$ 2, 2, 2, 2 $ | $2$ | $2$ | $(1,2)(3,8)(4,7)(5,6)$ | |
$ 4, 4 $ | $2$ | $4$ | $(1,2,8,3)(4,6,5,7)$ | |
$ 4, 4 $ | $2$ | $4$ | $(1,2,8,3)(4,7,5,6)$ | |
$ 2, 2, 2, 2 $ | $2$ | $2$ | $(1,4)(2,6)(3,7)(5,8)$ | |
$ 4, 4 $ | $2$ | $4$ | $(1,4,8,5)(2,6,3,7)$ | |
$ 2, 2, 2, 2 $ | $2$ | $2$ | $(1,4)(2,7)(3,6)(5,8)$ | |
$ 4, 4 $ | $2$ | $4$ | $(1,4,8,5)(2,7,3,6)$ | |
$ 2, 2, 2, 2 $ | $2$ | $2$ | $(1,6)(2,4)(3,5)(7,8)$ | |
$ 4, 4 $ | $2$ | $4$ | $(1,6,8,7)(2,4,3,5)$ | |
$ 4, 4 $ | $2$ | $4$ | $(1,6,8,7)(2,5,3,4)$ | |
$ 2, 2, 2, 2 $ | $2$ | $2$ | $(1,6)(2,5)(3,4)(7,8)$ | |
$ 2, 2, 2, 2 $ | $1$ | $2$ | $(1,8)(2,3)(4,5)(6,7)$ |
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: | $2$ | ||
Label: | 32.49 | magma: IdentifyGroup(G);
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Character table: |
1A | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 4A | 4B | 4C | 4D | 4E | 4F | ||
Size | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | |
2 P | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 2A | 2A | 2A | 2A | 2A | 2A | |
Type | ||||||||||||||||||
32.49.1a | R | |||||||||||||||||
32.49.1b | R | |||||||||||||||||
32.49.1c | R | |||||||||||||||||
32.49.1d | R | |||||||||||||||||
32.49.1e | R | |||||||||||||||||
32.49.1f | R | |||||||||||||||||
32.49.1g | R | |||||||||||||||||
32.49.1h | R | |||||||||||||||||
32.49.1i | R | |||||||||||||||||
32.49.1j | R | |||||||||||||||||
32.49.1k | R | |||||||||||||||||
32.49.1l | R | |||||||||||||||||
32.49.1m | R | |||||||||||||||||
32.49.1n | R | |||||||||||||||||
32.49.1o | R | |||||||||||||||||
32.49.1p | R | |||||||||||||||||
32.49.4a | R |
magma: CharacterTable(G);