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Magma
magma: G := TransitiveGroup(18, 10);
Group action invariants
Degree $n$: | $18$ | 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: | $C_3^2 : C_4$ | ||
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: | (1,13,10,15)(2,14,9,16)(3,5,7,18)(4,6,8,17)(11,12), (1,14)(2,13)(3,12)(4,11)(5,9)(6,10)(15,18)(16,17) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $4$: $C_4$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: None
Degree 6: $C_3^2:C_4$ x 2
Degree 9: $C_3^2:C_4$
Low degree siblings
6T10 x 2, 9T9, 12T17 x 2, 36T14Siblings 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 $ | $1$ | $1$ | $()$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $9$ | $2$ | $( 3,17)( 4,18)( 5,15)( 6,16)( 7,14)( 8,13)( 9,11)(10,12)$ |
$ 4, 4, 4, 4, 2 $ | $9$ | $4$ | $( 1, 2)( 3, 9,17,11)( 4,10,18,12)( 5, 7,15,14)( 6, 8,16,13)$ |
$ 4, 4, 4, 4, 2 $ | $9$ | $4$ | $( 1, 2)( 3,11,17, 9)( 4,12,18,10)( 5,14,15, 7)( 6,13,16, 8)$ |
$ 3, 3, 3, 3, 3, 3 $ | $4$ | $3$ | $( 1, 3,17)( 2, 4,18)( 5, 8, 9)( 6, 7,10)(11,13,15)(12,14,16)$ |
$ 3, 3, 3, 3, 3, 3 $ | $4$ | $3$ | $( 1, 6,16)( 2, 5,15)( 3, 7,12)( 4, 8,11)( 9,13,18)(10,14,17)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $36=2^{2} \cdot 3^{2}$ | 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: | 36.9 | magma: IdentifyGroup(G);
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Character table: |
2 2 2 2 2 . . 3 2 . . . 2 2 1a 2a 4a 4b 3a 3b 2P 1a 1a 2a 2a 3a 3b 3P 1a 2a 4b 4a 1a 1a X.1 1 1 1 1 1 1 X.2 1 1 -1 -1 1 1 X.3 1 -1 A -A 1 1 X.4 1 -1 -A A 1 1 X.5 4 . . . 1 -2 X.6 4 . . . -2 1 A = -E(4) = -Sqrt(-1) = -i |
magma: CharacterTable(G);