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Magma
magma: G := TransitiveGroup(18, 47);
Group action invariants
Degree $n$: | $18$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $47$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_3^2.A_4$ | ||
Parity: | $1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $6$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,13,9,4,15,11,5,18,8)(2,14,10,3,16,12,6,17,7), (1,17,9,4,14,11,5,16,8)(2,18,10,3,13,12,6,15,7) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $3$: $C_3$ x 4 $9$: $C_3^2$ $12$: $A_4$ $27$: $C_9:C_3$ $36$: $C_3\times A_4$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 3: $C_3$
Degree 6: $A_4$
Degree 9: $C_9:C_3$
Low degree siblings
18T47 x 2, 36T83Siblings 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, 1, 1, 1, 1, 1, 1 $ | $3$ | $2$ | $( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)$ |
$ 6, 6, 1, 1, 1, 1, 1, 1 $ | $3$ | $6$ | $( 7, 9,12, 8,10,11)(13,17,15,14,18,16)$ |
$ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ | $3$ | $3$ | $( 7,10,12)( 8, 9,11)(13,18,15)(14,17,16)$ |
$ 6, 6, 1, 1, 1, 1, 1, 1 $ | $3$ | $6$ | $( 7,11,10, 8,12, 9)(13,16,18,14,15,17)$ |
$ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ | $3$ | $3$ | $( 7,12,10)( 8,11, 9)(13,15,18)(14,16,17)$ |
$ 6, 3, 3, 2, 2, 2 $ | $3$ | $6$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 9,12, 8,10,11)(13,18,15)(14,17,16)$ |
$ 6, 3, 3, 2, 2, 2 $ | $3$ | $6$ | $( 1, 2)( 3, 4)( 5, 6)( 7,10,12)( 8, 9,11)(13,17,15,14,18,16)$ |
$ 6, 3, 3, 2, 2, 2 $ | $3$ | $6$ | $( 1, 2)( 3, 4)( 5, 6)( 7,11,10, 8,12, 9)(13,15,18)(14,16,17)$ |
$ 6, 3, 3, 2, 2, 2 $ | $3$ | $6$ | $( 1, 2)( 3, 4)( 5, 6)( 7,12,10)( 8,11, 9)(13,16,18,14,15,17)$ |
$ 6, 6, 3, 3 $ | $3$ | $6$ | $( 1, 3, 5, 2, 4, 6)( 7, 9,12, 8,10,11)(13,15,18)(14,16,17)$ |
$ 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 4, 5)( 2, 3, 6)( 7,10,12)( 8, 9,11)(13,15,18)(14,16,17)$ |
$ 6, 6, 3, 3 $ | $3$ | $6$ | $( 1, 5, 4)( 2, 6, 3)( 7,11,10, 8,12, 9)(13,17,15,14,18,16)$ |
$ 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 5, 4)( 2, 6, 3)( 7,12,10)( 8,11, 9)(13,18,15)(14,17,16)$ |
$ 9, 9 $ | $12$ | $9$ | $( 1, 7,13, 5,12,18, 4,10,15)( 2, 8,14, 6,11,17, 3, 9,16)$ |
$ 9, 9 $ | $12$ | $9$ | $( 1, 7,15, 5,12,13, 4,10,18)( 2, 8,16, 6,11,14, 3, 9,17)$ |
$ 9, 9 $ | $12$ | $9$ | $( 1, 7,17, 5,12,16, 4,10,14)( 2, 8,18, 6,11,15, 3, 9,13)$ |
$ 9, 9 $ | $12$ | $9$ | $( 1,13,10, 4,15,12, 5,18, 7)( 2,14, 9, 3,16,11, 6,17, 8)$ |
$ 9, 9 $ | $12$ | $9$ | $( 1,13, 8, 4,15, 9, 5,18,11)( 2,14, 7, 3,16,10, 6,17,12)$ |
$ 9, 9 $ | $12$ | $9$ | $( 1,13,12, 4,15, 7, 5,18,10)( 2,14,11, 3,16, 8, 6,17, 9)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $108=2^{2} \cdot 3^{3}$ | 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: | 108.21 | magma: IdentifyGroup(G);
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Character table: |
2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 . . . . . . 3 3 2 2 2 2 2 2 2 2 2 2 3 2 3 2 2 2 2 2 2 1a 2a 6a 3a 6b 3b 6c 6d 6e 6f 6g 3c 6h 3d 9a 9b 9c 9d 9e 9f 2P 1a 1a 3b 3b 3a 3a 3b 3b 3a 3a 3d 3d 3c 3c 9f 9e 9d 9c 9b 9a 3P 1a 2a 2a 1a 2a 1a 2a 2a 2a 2a 2a 1a 2a 1a 3d 3d 3d 3c 3c 3c 5P 1a 2a 6b 3b 6a 3a 6e 6f 6c 6d 6h 3d 6g 3c 9f 9e 9d 9c 9b 9a 7P 1a 2a 6a 3a 6b 3b 6c 6d 6e 6f 6g 3c 6h 3d 9a 9b 9c 9d 9e 9f X.1 1 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 A A A /A /A /A X.3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 /A /A /A A A A X.4 1 1 A A /A /A A A /A /A 1 1 1 1 1 A /A A /A 1 X.5 1 1 /A /A A A /A /A A A 1 1 1 1 1 /A A /A A 1 X.6 1 1 A A /A /A A A /A /A 1 1 1 1 A /A 1 1 A /A X.7 1 1 /A /A A A /A /A A A 1 1 1 1 /A A 1 1 /A A X.8 1 1 A A /A /A A A /A /A 1 1 1 1 /A 1 A /A 1 A X.9 1 1 /A /A A A /A /A A A 1 1 1 1 A 1 /A A 1 /A X.10 3 -1 -1 3 -1 3 -1 -1 -1 -1 -1 3 -1 3 . . . . . . X.11 3 3 . . . . . . . . /C /C C C . . . . . . X.12 3 3 . . . . . . . . C C /C /C . . . . . . X.13 3 -1 -/A C -A /C -/A -/A -A -A -1 3 -1 3 . . . . . . X.14 3 -1 -A /C -/A C -A -A -/A -/A -1 3 -1 3 . . . . . . X.15 3 -1 B . /B . /B 2 B 2 -A /C -/A C . . . . . . X.16 3 -1 /B . B . B 2 /B 2 -/A C -A /C . . . . . . X.17 3 -1 2 . 2 . B /B /B B -A /C -/A C . . . . . . X.18 3 -1 2 . 2 . /B B B /B -/A C -A /C . . . . . . X.19 3 -1 /B . B . 2 B 2 /B -A /C -/A C . . . . . . X.20 3 -1 B . /B . 2 /B 2 B -/A C -A /C . . . . . . A = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 B = 2*E(3) = -1+Sqrt(-3) = 2b3 C = 3*E(3) = (-3+3*Sqrt(-3))/2 = 3b3 |
magma: CharacterTable(G);