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Magma
magma: G := TransitiveGroup(18, 22);
Group action invariants
Degree $n$: | $18$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $22$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_3^2:C_6$ | ||
Parity: | $-1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $3$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,10,8,18,14,5)(2,12,9,17,15,4)(3,11,7,16,13,6), (1,18,9,6,13,12)(2,17,7,5,14,11)(3,16,8,4,15,10) | 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 2: $C_2$
Degree 3: None
Degree 6: $S_3\times C_3$
Degree 9: None
Low degree siblings
9T11, 9T13, 18T20, 18T21, 27T11Siblings 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, 3, 3, 1, 1, 1 $ | $6$ | $3$ | $( 4,12,18)( 5,10,16)( 6,11,17)( 7, 8, 9)(13,15,14)$ |
$ 3, 3, 3, 3, 3, 1, 1, 1 $ | $6$ | $3$ | $( 4,18,12)( 5,16,10)( 6,17,11)( 7, 9, 8)(13,14,15)$ |
$ 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $9$ | $2$ | $( 1, 4)( 2, 6)( 3, 5)( 7,12)( 8,11)( 9,10)(13,16)(14,18)(15,17)$ |
$ 6, 6, 6 $ | $9$ | $6$ | $( 1, 4, 7,11,15,18)( 2, 6, 8,10,13,17)( 3, 5, 9,12,14,16)$ |
$ 6, 6, 6 $ | $9$ | $6$ | $( 1, 4,14,17, 8,12)( 2, 6,15,16, 9,11)( 3, 5,13,18, 7,10)$ |
$ 3, 3, 3, 3, 3, 3 $ | $3$ | $3$ | $( 1, 7,15)( 2, 8,13)( 3, 9,14)( 4,11,18)( 5,12,16)( 6,10,17)$ |
$ 3, 3, 3, 3, 3, 3 $ | $6$ | $3$ | $( 1, 7,13)( 2, 8,14)( 3, 9,15)( 4,16,12)( 5,17,10)( 6,18,11)$ |
$ 3, 3, 3, 3, 3, 3 $ | $3$ | $3$ | $( 1,13, 9)( 2,14, 7)( 3,15, 8)( 4,16,10)( 5,17,11)( 6,18,12)$ |
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: | 54.5 | magma: IdentifyGroup(G);
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Character table: |
2 1 . . . 1 1 1 1 . 1 3 3 2 2 3 1 1 1 2 2 2 1a 3a 3b 3c 2a 6a 6b 3d 3e 3f 2P 1a 3b 3a 3c 1a 3d 3f 3f 3e 3d 3P 1a 1a 1a 1a 2a 2a 2a 1a 1a 1a 5P 1a 3b 3a 3c 2a 6b 6a 3f 3e 3d X.1 1 1 1 1 1 1 1 1 1 1 X.2 1 1 1 1 -1 -1 -1 1 1 1 X.3 1 A /A 1 -1 -A -/A /A 1 A X.4 1 /A A 1 -1 -/A -A A 1 /A X.5 1 A /A 1 1 A /A /A 1 A X.6 1 /A A 1 1 /A A A 1 /A X.7 2 -1 -1 2 . . . 2 -1 2 X.8 2 -A -/A 2 . . . B -1 /B X.9 2 -/A -A 2 . . . /B -1 B X.10 6 . . -3 . . . . . . A = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 B = 2*E(3) = -1+Sqrt(-3) = 2b3 |
magma: CharacterTable(G);