Show commands:
Magma
magma: G := TransitiveGroup(9, 10);
Group action invariants
Degree $n$: | $9$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $10$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $(C_9:C_3):C_2$ | ||
CHM label: | $[3^{2}]S(3)_{6}$ | ||
Parity: | $1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $1$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,4,7)(2,8,5), (1,2,3,4,5,6,7,8,9), (1,8)(2,7)(3,6)(4,5) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $3$: $C_3$ $6$: $S_3$, $C_6$ $18$: $S_3\times C_3$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $S_3$
Low degree siblings
18T18, 27T14Siblings 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 $ | $1$ | $1$ | $()$ | |
$ 6, 2, 1 $ | $9$ | $6$ | $(2,3,5,9,8,6)(4,7)$ | |
$ 3, 3, 1, 1, 1 $ | $3$ | $3$ | $(2,5,8)(3,9,6)$ | |
$ 6, 2, 1 $ | $9$ | $6$ | $(2,6,8,9,5,3)(4,7)$ | |
$ 3, 3, 1, 1, 1 $ | $3$ | $3$ | $(2,8,5)(3,6,9)$ | |
$ 2, 2, 2, 2, 1 $ | $9$ | $2$ | $(2,9)(3,8)(4,7)(5,6)$ | |
$ 9 $ | $6$ | $9$ | $(1,2,3,4,5,6,7,8,9)$ | |
$ 9 $ | $6$ | $9$ | $(1,2,6,4,5,9,7,8,3)$ | |
$ 9 $ | $6$ | $9$ | $(1,2,9,4,5,3,7,8,6)$ | |
$ 3, 3, 3 $ | $2$ | $3$ | $(1,4,7)(2,5,8)(3,6,9)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $54=2 \cdot 3^{3}$ | 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: | not nilpotent | ||
Label: | 54.6 | magma: IdentifyGroup(G);
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Character table: |
1A | 2A | 3A | 3B1 | 3B-1 | 6A1 | 6A-1 | 9A | 9B1 | 9B-1 | ||
Size | 1 | 9 | 2 | 3 | 3 | 9 | 9 | 6 | 6 | 6 | |
2 P | 1A | 1A | 3A | 3B-1 | 3B1 | 3B1 | 3B-1 | 9A | 9B-1 | 9B1 | |
3 P | 1A | 2A | 1A | 1A | 1A | 2A | 2A | 3A | 3A | 3A | |
Type | |||||||||||
54.6.1a | R | ||||||||||
54.6.1b | R | ||||||||||
54.6.1c1 | C | ||||||||||
54.6.1c2 | C | ||||||||||
54.6.1d1 | C | ||||||||||
54.6.1d2 | C | ||||||||||
54.6.2a | R | ||||||||||
54.6.2b1 | C | ||||||||||
54.6.2b2 | C | ||||||||||
54.6.6a | R |
magma: CharacterTable(G);