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Magma
magma: G := TransitiveGroup(27, 20);
Group action invariants
Degree $n$: | $27$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $20$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_3.\He_3$ | ||
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: | (4,5,6)(7,15,10)(8,13,11)(9,14,12)(16,20,22)(17,21,23)(18,19,24)(25,27,26), (1,12,20,3,11,19,2,10,21)(4,13,23,6,15,22,5,14,24)(7,16,26,9,18,25,8,17,27) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $3$: $C_3$ x 4 $9$: $C_3^2$ $27$: $C_3^2:C_3$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $C_3$
Degree 9: $C_3^2:C_3$
Low degree siblings
27T26Siblings 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 $ | $1$ | $1$ | $()$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1 $ | $9$ | $3$ | $( 4, 5, 6)( 7,15,10)( 8,13,11)( 9,14,12)(16,20,22)(17,21,23)(18,19,24) (25,27,26)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1 $ | $9$ | $3$ | $( 4, 6, 5)( 7,10,15)( 8,11,13)( 9,12,14)(16,22,20)(17,23,21)(18,24,19) (25,26,27)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21) (22,23,24)(25,26,27)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,12,11)(13,15,14)(16,18,17)(19,21,20) (22,24,23)(25,27,26)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $3$ | $3$ | $( 1, 4,25)( 2, 5,26)( 3, 6,27)( 7,11,15)( 8,12,13)( 9,10,14)(16,19,22) (17,20,23)(18,21,24)$ |
$ 9, 9, 9 $ | $9$ | $9$ | $( 1, 7,17, 2, 8,18, 3, 9,16)( 4,10,21, 5,11,19, 6,12,20)(13,22,26,14,23,27,15, 24,25)$ |
$ 9, 9, 9 $ | $3$ | $9$ | $( 1, 7,24, 3, 9,23, 2, 8,22)( 4,11,18, 6,10,17, 5,12,16)(13,19,25,15,21,27,14, 20,26)$ |
$ 9, 9, 9 $ | $9$ | $9$ | $( 1, 7,21, 2, 8,19, 3, 9,20)( 4,12,23, 5,10,24, 6,11,22)(13,18,25,14,16,26,15, 17,27)$ |
$ 9, 9, 9 $ | $3$ | $9$ | $( 1, 8,23, 3, 7,22, 2, 9,24)( 4,12,17, 6,11,16, 5,10,18)(13,20,27,15,19,26,14, 21,25)$ |
$ 9, 9, 9 $ | $3$ | $9$ | $( 1, 9,22, 3, 8,24, 2, 7,23)( 4,10,16, 6,12,18, 5,11,17)(13,21,26,15,20,25,14, 19,27)$ |
$ 9, 9, 9 $ | $3$ | $9$ | $( 1,16,13, 2,17,14, 3,18,15)( 4,19, 8, 5,20, 9, 6,21, 7)(10,27,24,11,25,22,12, 26,23)$ |
$ 9, 9, 9 $ | $9$ | $9$ | $( 1,16, 9, 3,18, 8, 2,17, 7)( 4,20,12, 6,19,11, 5,21,10)(13,25,24,15,27,23,14, 26,22)$ |
$ 9, 9, 9 $ | $9$ | $9$ | $( 1,16,12, 3,18,11, 2,17,10)( 4,21,14, 6,20,13, 5,19,15)( 7,27,22, 9,26,24, 8, 25,23)$ |
$ 9, 9, 9 $ | $3$ | $9$ | $( 1,17,15, 2,18,13, 3,16,14)( 4,20, 7, 5,21, 8, 6,19, 9)(10,25,23,11,26,24,12, 27,22)$ |
$ 9, 9, 9 $ | $3$ | $9$ | $( 1,18,14, 2,16,15, 3,17,13)( 4,21, 9, 5,19, 7, 6,20, 8)(10,26,22,11,27,23,12, 25,24)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $3$ | $3$ | $( 1,25, 4)( 2,26, 5)( 3,27, 6)( 7,15,11)( 8,13,12)( 9,14,10)(16,22,19) (17,23,20)(18,24,21)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $81=3^{4}$ | 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: | $3$ | ||
Label: | 81.8 | magma: IdentifyGroup(G);
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Character table: |
3 4 2 2 4 4 3 2 3 2 3 3 3 2 2 3 3 3 1a 3a 3b 3c 3d 3e 9a 9b 9c 9d 9e 9f 9g 9h 9i 9j 3f 2P 1a 3b 3a 3d 3c 3f 9g 9j 9h 9i 9f 9d 9a 9c 9b 9e 3e 3P 1a 1a 1a 1a 1a 1a 3c 3d 3c 3d 3d 3c 3d 3d 3c 3c 1a 5P 1a 3b 3a 3d 3c 3f 9g 9i 9h 9f 9j 9e 9a 9c 9d 9b 3e 7P 1a 3a 3b 3c 3d 3e 9a 9d 9c 9e 9b 9j 9g 9h 9f 9i 3f X.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 A A A A A /A /A /A /A /A 1 X.3 1 1 1 1 1 1 /A /A /A /A /A A A A A A 1 X.4 1 A /A 1 1 1 1 A /A A A /A 1 A /A /A 1 X.5 1 /A A 1 1 1 1 /A A /A /A A 1 /A A A 1 X.6 1 A /A 1 1 1 A /A 1 /A /A A /A 1 A A 1 X.7 1 /A A 1 1 1 /A A 1 A A /A A 1 /A /A 1 X.8 1 A /A 1 1 1 /A 1 A 1 1 1 A /A 1 1 1 X.9 1 /A A 1 1 1 A 1 /A 1 1 1 /A A 1 1 1 X.10 3 . . 3 3 B . . . . . . . . . . /B X.11 3 . . 3 3 /B . . . . . . . . . . B X.12 3 . . B /B . . C . D E /C . . /E /D . X.13 3 . . B /B . . D . E C /D . . /C /E . X.14 3 . . B /B . . E . C D /E . . /D /C . X.15 3 . . /B B . . /D . /E /C D . . C E . X.16 3 . . /B B . . /C . /D /E C . . E D . X.17 3 . . /B B . . /E . /C /D E . . D C . A = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 B = 3*E(3)^2 = (-3-3*Sqrt(-3))/2 = -3-3b3 C = -E(9)^4-2*E(9)^7 D = -E(9)^4+E(9)^7 E = 2*E(9)^4+E(9)^7 |
magma: CharacterTable(G);