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Magma
magma: G := TransitiveGroup(18, 19);
Group action invariants
Degree $n$: | $18$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $19$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_3\times D_9$ | ||
Parity: | $-1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $9$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,5,7,2,6,8,3,4,9)(10,14,18,12,13,17,11,15,16), (1,15,2,13,3,14)(4,12,5,10,6,11)(7,18,8,16,9,17) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $3$: $C_3$ $6$: $S_3$, $C_6$ $18$: $S_3\times C_3$, $D_{9}$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: $S_3$
Degree 6: $S_3$
Degree 9: None
Low degree siblings
27T9Siblings 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 $ | $1$ | $1$ | $()$ |
$ 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $2$ | $3$ | $(10,11,12)(13,14,15)(16,17,18)$ |
$ 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $2$ | $3$ | $(10,12,11)(13,15,14)(16,18,17)$ |
$ 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)$ |
$ 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,12,11)(13,15,14)(16,18,17)$ |
$ 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,12,11)(13,15,14)(16,18,17)$ |
$ 9, 9 $ | $2$ | $9$ | $( 1, 4, 8, 2, 5, 9, 3, 6, 7)(10,13,16,12,15,18,11,14,17)$ |
$ 9, 9 $ | $2$ | $9$ | $( 1, 4, 8, 2, 5, 9, 3, 6, 7)(10,14,18,12,13,17,11,15,16)$ |
$ 9, 9 $ | $2$ | $9$ | $( 1, 4, 8, 2, 5, 9, 3, 6, 7)(10,15,17,12,14,16,11,13,18)$ |
$ 9, 9 $ | $2$ | $9$ | $( 1, 5, 7, 2, 6, 8, 3, 4, 9)(10,13,16,12,15,18,11,14,17)$ |
$ 9, 9 $ | $2$ | $9$ | $( 1, 5, 7, 2, 6, 8, 3, 4, 9)(10,14,18,12,13,17,11,15,16)$ |
$ 9, 9 $ | $2$ | $9$ | $( 1, 5, 7, 2, 6, 8, 3, 4, 9)(10,15,17,12,14,16,11,13,18)$ |
$ 9, 9 $ | $2$ | $9$ | $( 1, 6, 9, 2, 4, 7, 3, 5, 8)(10,13,16,12,15,18,11,14,17)$ |
$ 9, 9 $ | $2$ | $9$ | $( 1, 6, 9, 2, 4, 7, 3, 5, 8)(10,14,18,12,13,17,11,15,16)$ |
$ 9, 9 $ | $2$ | $9$ | $( 1, 6, 9, 2, 4, 7, 3, 5, 8)(10,15,17,12,14,16,11,13,18)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $9$ | $2$ | $( 1,10)( 2,11)( 3,12)( 4,17)( 5,18)( 6,16)( 7,13)( 8,14)( 9,15)$ |
$ 6, 6, 6 $ | $9$ | $6$ | $( 1,10, 2,11, 3,12)( 4,17, 5,18, 6,16)( 7,13, 8,14, 9,15)$ |
$ 6, 6, 6 $ | $9$ | $6$ | $( 1,10, 3,12, 2,11)( 4,17, 6,16, 5,18)( 7,13, 9,15, 8,14)$ |
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: | 54.3 | magma: IdentifyGroup(G);
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Character table: |
2 1 . . 1 . 1 . . . . . . . . . 1 1 1 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 1 1 1 1a 3a 3b 3c 3d 3e 9a 9b 9c 9d 9e 9f 9g 9h 9i 2a 6a 6b 2P 1a 3b 3a 3e 3d 3c 9h 9e 9b 9g 9d 9a 9i 9f 9c 1a 3c 3e 3P 1a 1a 1a 1a 1a 1a 3d 3d 3d 3d 3d 3d 3d 3d 3d 2a 2a 2a 5P 1a 3b 3a 3e 3d 3c 9f 9c 9i 9e 9b 9h 9d 9a 9g 2a 6b 6a 7P 1a 3a 3b 3c 3d 3e 9h 9i 9g 9b 9c 9a 9e 9f 9d 2a 6a 6b X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 X.3 1 A /A /A 1 A 1 A /A A /A 1 /A 1 A -1 -A -/A X.4 1 /A A A 1 /A 1 /A A /A A 1 A 1 /A -1 -/A -A X.5 1 A /A /A 1 A 1 A /A A /A 1 /A 1 A 1 A /A X.6 1 /A A A 1 /A 1 /A A /A A 1 A 1 /A 1 /A A X.7 2 2 2 2 2 2 -1 -1 -1 -1 -1 -1 -1 -1 -1 . . . X.8 2 B /B /B 2 B -1 -/A -A -/A -A -1 -A -1 -/A . . . X.9 2 /B B B 2 /B -1 -A -/A -A -/A -1 -/A -1 -A . . . X.10 2 -/A -A /B -1 B C F /H H /G E /F D G . . . X.11 2 -/A -A /B -1 B D G /F F /H C /G E H . . . X.12 2 -/A -A /B -1 B E H /G G /F D /H C F . . . X.13 2 -A -/A B -1 /B C /F H /H G E F D /G . . . X.14 2 -A -/A B -1 /B D /G F /F H C G E /H . . . X.15 2 -A -/A B -1 /B E /H G /G F D H C /F . . . X.16 2 -1 -1 2 -1 2 C E D D C E E D C . . . X.17 2 -1 -1 2 -1 2 D C E E D C C E D . . . X.18 2 -1 -1 2 -1 2 E D C C E D D C E . . . A = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 B = 2*E(3) = -1+Sqrt(-3) = 2b3 C = E(9)^2+E(9)^7 D = E(9)^4+E(9)^5 E = -E(9)^2-E(9)^4-E(9)^5-E(9)^7 F = E(9)^2+E(9)^4 G = -E(9)^4+E(9)^5-E(9)^7 H = -E(9)^2-E(9)^5+E(9)^7 |
magma: CharacterTable(G);