Show commands:
Magma
magma: G := TransitiveGroup(6, 16);
Group action invariants
Degree $n$: | $6$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $16$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $S_6$ | ||
CHM label: | $S6$ | ||
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: | (1,2), (1,2,3,4,5,6) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 3: None
Low degree siblings
6T16, 10T32, 12T183 x 2, 15T28 x 2, 20T145, 20T149 x 2, 30T164 x 2, 30T166 x 2, 30T176 x 2, 36T1252, 40T589, 40T592 x 2, 45T96Siblings 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 | Index | Representative |
1A | $1^{6}$ | $1$ | $1$ | $0$ | $()$ |
2A | $2^{3}$ | $15$ | $2$ | $3$ | $(1,2)(3,4)(5,6)$ |
2B | $2,1^{4}$ | $15$ | $2$ | $1$ | $(1,2)$ |
2C | $2^{2},1^{2}$ | $45$ | $2$ | $2$ | $(1,2)(3,4)$ |
3A | $3^{2}$ | $40$ | $3$ | $4$ | $(1,2,3)(4,5,6)$ |
3B | $3,1^{3}$ | $40$ | $3$ | $2$ | $(1,2,3)$ |
4A | $4,2$ | $90$ | $4$ | $4$ | $(1,2,3,4)(5,6)$ |
4B | $4,1^{2}$ | $90$ | $4$ | $3$ | $(1,2,3,4)$ |
5A | $5,1$ | $144$ | $5$ | $4$ | $(1,2,3,4,5)$ |
6A | $6$ | $120$ | $6$ | $5$ | $(1,2,3,4,5,6)$ |
6B | $3,2,1$ | $120$ | $6$ | $3$ | $(1,2,3)(4,5)$ |
magma: ConjugacyClasses(G);
Malle's constant $a(G)$: $1$
Group invariants
Order: | $720=2^{4} \cdot 3^{2} \cdot 5$ | 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: | no | magma: IsSolvable(G);
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Nilpotency class: | not nilpotent | ||
Label: | 720.763 | magma: IdentifyGroup(G);
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Character table: |
1A | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 5A | 6A | 6B | ||
Size | 1 | 15 | 15 | 45 | 40 | 40 | 90 | 90 | 144 | 120 | 120 | |
2 P | 1A | 1A | 1A | 1A | 3A | 3B | 2C | 2C | 5A | 3A | 3B | |
3 P | 1A | 2A | 2B | 2C | 1A | 1A | 4A | 4B | 5A | 2A | 2B | |
5 P | 1A | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 1A | 6A | 6B | |
Type | ||||||||||||
720.763.1a | R | |||||||||||
720.763.1b | R | |||||||||||
720.763.5a | R | |||||||||||
720.763.5b | R | |||||||||||
720.763.5c | R | |||||||||||
720.763.5d | R | |||||||||||
720.763.9a | R | |||||||||||
720.763.9b | R | |||||||||||
720.763.10a | R | |||||||||||
720.763.10b | R | |||||||||||
720.763.16a | R |
magma: CharacterTable(G);