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Magma
magma: G := TransitiveGroup(27, 9);
Group action invariants
Degree $n$: | $27$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $9$ | 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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$\card{\Aut(F/K)}$: | $3$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,22,17,11,6,25,20,13,7)(2,23,18,12,4,26,21,14,8)(3,24,16,10,5,27,19,15,9), (1,2,3)(4,27,6,26,5,25)(7,23,9,22,8,24)(10,20,12,19,11,21)(13,18,15,17,14,16) | 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 9: $D_{9}$, $S_3\times C_3$
Low degree siblings
18T19Siblings 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, 1, 1, 1, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1 $ | $9$ | $2$ | $( 4,26)( 5,27)( 6,25)( 7,22)( 8,23)( 9,24)(10,19)(11,20)(12,21)(13,17)(14,18) (15,16)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21) (22,23,24)(25,26,27)$ |
$ 6, 6, 6, 6, 3 $ | $9$ | $6$ | $( 1, 2, 3)( 4,27, 6,26, 5,25)( 7,23, 9,22, 8,24)(10,20,12,19,11,21) (13,18,15,17,14,16)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,12,11)(13,15,14)(16,18,17)(19,21,20) (22,24,23)(25,27,26)$ |
$ 6, 6, 6, 6, 3 $ | $9$ | $6$ | $( 1, 3, 2)( 4,25, 5,26, 6,27)( 7,24, 8,22, 9,23)(10,21,11,19,12,20) (13,16,14,17,15,18)$ |
$ 9, 9, 9 $ | $2$ | $9$ | $( 1, 4, 9,11,14,16,20,23,27)( 2, 5, 7,12,15,17,21,24,25)( 3, 6, 8,10,13,18,19, 22,26)$ |
$ 9, 9, 9 $ | $2$ | $9$ | $( 1, 5, 8,11,15,18,20,24,26)( 2, 6, 9,12,13,16,21,22,27)( 3, 4, 7,10,14,17,19, 23,25)$ |
$ 9, 9, 9 $ | $2$ | $9$ | $( 1, 6, 7,11,13,17,20,22,25)( 2, 4, 8,12,14,18,21,23,26)( 3, 5, 9,10,15,16,19, 24,27)$ |
$ 9, 9, 9 $ | $2$ | $9$ | $( 1, 7,13,20,25, 6,11,17,22)( 2, 8,14,21,26, 4,12,18,23)( 3, 9,15,19,27, 5,10, 16,24)$ |
$ 9, 9, 9 $ | $2$ | $9$ | $( 1, 8,15,20,26, 5,11,18,24)( 2, 9,13,21,27, 6,12,16,22)( 3, 7,14,19,25, 4,10, 17,23)$ |
$ 9, 9, 9 $ | $2$ | $9$ | $( 1, 9,14,20,27, 4,11,16,23)( 2, 7,15,21,25, 5,12,17,24)( 3, 8,13,19,26, 6,10, 18,22)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1,10,21)( 2,11,19)( 3,12,20)( 4,13,24)( 5,14,22)( 6,15,23)( 7,16,26) ( 8,17,27)( 9,18,25)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1,11,20)( 2,12,21)( 3,10,19)( 4,14,23)( 5,15,24)( 6,13,22)( 7,17,25) ( 8,18,26)( 9,16,27)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1,12,19)( 2,10,20)( 3,11,21)( 4,15,22)( 5,13,23)( 6,14,24)( 7,18,27) ( 8,16,25)( 9,17,26)$ |
$ 9, 9, 9 $ | $2$ | $9$ | $( 1,13,25,11,22, 7,20, 6,17)( 2,14,26,12,23, 8,21, 4,18)( 3,15,27,10,24, 9,19, 5,16)$ |
$ 9, 9, 9 $ | $2$ | $9$ | $( 1,14,27,11,23, 9,20, 4,16)( 2,15,25,12,24, 7,21, 5,17)( 3,13,26,10,22, 8,19, 6,18)$ |
$ 9, 9, 9 $ | $2$ | $9$ | $( 1,15,26,11,24, 8,20, 5,18)( 2,13,27,12,22, 9,21, 6,16)( 3,14,25,10,23, 7,19, 4,17)$ |
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 1 3 1 3 1 3 3 3 3 3 3 3 3 3 3 3 3 1a 2a 3a 6a 3b 6b 9a 9b 9c 9d 9e 9f 3c 3d 3e 9g 9h 9i 2P 1a 1a 3b 3b 3a 3a 9f 9e 9d 9g 9i 9h 3e 3d 3c 9c 9b 9a 3P 1a 2a 1a 2a 1a 2a 3d 3d 3d 3d 3d 3d 1a 1a 1a 3d 3d 3d 5P 1a 2a 3b 6b 3a 6a 9i 9h 9g 9c 9b 9a 3e 3d 3c 9d 9f 9e 7P 1a 2a 3a 6a 3b 6b 9e 9f 9d 9g 9h 9i 3c 3d 3e 9c 9a 9b 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 -1 A -A /A -/A A /A 1 1 A /A /A 1 A 1 A /A X.4 1 -1 /A -/A A -A /A A 1 1 /A A A 1 /A 1 /A A X.5 1 1 A A /A /A A /A 1 1 A /A /A 1 A 1 A /A X.6 1 1 /A /A A A /A A 1 1 /A A A 1 /A 1 /A A X.7 2 . 2 . 2 . -1 -1 -1 -1 -1 -1 2 2 2 -1 -1 -1 X.8 2 . B . /B . -A -/A -1 -1 -A -/A /B 2 B -1 -A -/A X.9 2 . /B . B . -/A -A -1 -1 -/A -A B 2 /B -1 -/A -A X.10 2 . 2 . 2 . C C C D D D -1 -1 -1 E E E X.11 2 . 2 . 2 . D D D E E E -1 -1 -1 C C C X.12 2 . 2 . 2 . E E E C C C -1 -1 -1 D D D X.13 2 . B . /B . F /F E C G /G -/A -1 -A D H /H X.14 2 . B . /B . G /G C D H /H -/A -1 -A E F /F X.15 2 . B . /B . H /H D E F /F -/A -1 -A C G /G X.16 2 . /B . B . /H H D E /F F -A -1 -/A C /G G X.17 2 . /B . B . /F F E C /G G -A -1 -/A D /H H X.18 2 . /B . B . /G G C D /H H -A -1 -/A E /F F A = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 B = 2*E(3)^2 = -1-Sqrt(-3) = -1-i3 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)^5+E(9)^7 G = -E(9)^2+E(9)^4-E(9)^5 H = E(9)^2-E(9)^4-E(9)^7 |
magma: CharacterTable(G);