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Magma
magma: G := TransitiveGroup(18, 28);
Group action invariants
Degree $n$: | $18$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $28$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $F_9$ | ||
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: | (1,8,14)(2,7,13)(3,9,16)(4,10,15)(5,12,18)(6,11,17), (1,12,3,18,6,13,9,7)(2,11,4,17,5,14,10,8)(15,16) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $4$: $C_4$ $8$: $C_8$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: None
Degree 6: None
Degree 9: $C_3^2:C_8$
Low degree siblings
9T15, 12T46, 24T81, 36T49Siblings 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, 1, 1, 1, 1, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ |
$ 4, 4, 4, 4, 1, 1 $ | $9$ | $4$ | $( 3, 9,17,11)( 4,10,18,12)( 5, 7,15,13)( 6, 8,16,14)$ |
$ 4, 4, 4, 4, 1, 1 $ | $9$ | $4$ | $( 3,11,17, 9)( 4,12,18,10)( 5,13,15, 7)( 6,14,16, 8)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $9$ | $2$ | $( 3,17)( 4,18)( 5,15)( 6,16)( 7,13)( 8,14)( 9,11)(10,12)$ |
$ 8, 8, 2 $ | $9$ | $8$ | $( 1, 2)( 3, 5,11,13,17,15, 9, 7)( 4, 6,12,14,18,16,10, 8)$ |
$ 8, 8, 2 $ | $9$ | $8$ | $( 1, 2)( 3, 7, 9,15,17,13,11, 5)( 4, 8,10,16,18,14,12, 6)$ |
$ 8, 8, 2 $ | $9$ | $8$ | $( 1, 2)( 3,13, 9, 5,17, 7,11,15)( 4,14,10, 6,18, 8,12,16)$ |
$ 8, 8, 2 $ | $9$ | $8$ | $( 1, 2)( 3,15,11, 7,17, 5, 9,13)( 4,16,12, 8,18, 6,10,14)$ |
$ 3, 3, 3, 3, 3, 3 $ | $8$ | $3$ | $( 1, 3,17)( 2, 4,18)( 5, 7,10)( 6, 8, 9)(11,14,16)(12,13,15)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $72=2^{3} \cdot 3^{2}$ | 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: | 72.39 | magma: IdentifyGroup(G);
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Character table: |
2 3 3 3 3 3 3 3 3 . 3 2 . . . . . . . 2 1a 4a 4b 2a 8a 8b 8c 8d 3a 2P 1a 2a 2a 1a 4b 4a 4a 4b 3a 3P 1a 4b 4a 2a 8c 8d 8a 8b 1a 5P 1a 4a 4b 2a 8d 8c 8b 8a 3a 7P 1a 4b 4a 2a 8b 8a 8d 8c 3a X.1 1 1 1 1 1 1 1 1 1 X.2 1 1 1 1 -1 -1 -1 -1 1 X.3 1 -1 -1 1 A -A -A A 1 X.4 1 -1 -1 1 -A A A -A 1 X.5 1 A -A -1 B /B -/B -B 1 X.6 1 A -A -1 -B -/B /B B 1 X.7 1 -A A -1 -/B -B B /B 1 X.8 1 -A A -1 /B B -B -/B 1 X.9 8 . . . . . . . -1 A = -E(4) = -Sqrt(-1) = -i B = -E(8) |
magma: CharacterTable(G);