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Magma
magma: G := TransitiveGroup(27, 2);
Group action invariants
| Degree $n$: | $27$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $2$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_3\times C_9$ | ||
| Parity: | $1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $27$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,6,9,12,14,16,19,23,26)(2,4,7,10,15,17,20,24,27)(3,5,8,11,13,18,21,22,25), (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) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $3$: $C_3$ x 4 $9$: $C_9$ x 3, $C_3^2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $C_3$ x 4
Low degree siblings
There are no siblings 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, 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 $ | $1$ | $3$ | $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,12,11)(13,15,14)(16,18,17)(19,21,20) (22,24,23)(25,27,26)$ |
| $ 9, 9, 9 $ | $1$ | $9$ | $( 1, 4, 8,12,15,18,19,24,25)( 2, 5, 9,10,13,16,20,22,26)( 3, 6, 7,11,14,17,21, 23,27)$ |
| $ 9, 9, 9 $ | $1$ | $9$ | $( 1, 5, 7,12,13,17,19,22,27)( 2, 6, 8,10,14,18,20,23,25)( 3, 4, 9,11,15,16,21, 24,26)$ |
| $ 9, 9, 9 $ | $1$ | $9$ | $( 1, 6, 9,12,14,16,19,23,26)( 2, 4, 7,10,15,17,20,24,27)( 3, 5, 8,11,13,18,21, 22,25)$ |
| $ 9, 9, 9 $ | $1$ | $9$ | $( 1, 7,13,19,27, 5,12,17,22)( 2, 8,14,20,25, 6,10,18,23)( 3, 9,15,21,26, 4,11, 16,24)$ |
| $ 9, 9, 9 $ | $1$ | $9$ | $( 1, 8,15,19,25, 4,12,18,24)( 2, 9,13,20,26, 5,10,16,22)( 3, 7,14,21,27, 6,11, 17,23)$ |
| $ 9, 9, 9 $ | $1$ | $9$ | $( 1, 9,14,19,26, 6,12,16,23)( 2, 7,15,20,27, 4,10,17,24)( 3, 8,13,21,25, 5,11, 18,22)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1,10,21)( 2,11,19)( 3,12,20)( 4,13,23)( 5,14,24)( 6,15,22)( 7,18,26) ( 8,16,27)( 9,17,25)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1,11,20)( 2,12,21)( 3,10,19)( 4,14,22)( 5,15,23)( 6,13,24)( 7,16,25) ( 8,17,26)( 9,18,27)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1,12,19)( 2,10,20)( 3,11,21)( 4,15,24)( 5,13,22)( 6,14,23)( 7,17,27) ( 8,18,25)( 9,16,26)$ |
| $ 9, 9, 9 $ | $1$ | $9$ | $( 1,13,27,12,22, 7,19, 5,17)( 2,14,25,10,23, 8,20, 6,18)( 3,15,26,11,24, 9,21, 4,16)$ |
| $ 9, 9, 9 $ | $1$ | $9$ | $( 1,14,26,12,23, 9,19, 6,16)( 2,15,27,10,24, 7,20, 4,17)( 3,13,25,11,22, 8,21, 5,18)$ |
| $ 9, 9, 9 $ | $1$ | $9$ | $( 1,15,25,12,24, 8,19, 4,18)( 2,13,26,10,22, 9,20, 5,16)( 3,14,27,11,23, 7,21, 6,17)$ |
| $ 9, 9, 9 $ | $1$ | $9$ | $( 1,16, 6,19, 9,23,12,26,14)( 2,17, 4,20, 7,24,10,27,15)( 3,18, 5,21, 8,22,11, 25,13)$ |
| $ 9, 9, 9 $ | $1$ | $9$ | $( 1,17, 5,19, 7,22,12,27,13)( 2,18, 6,20, 8,23,10,25,14)( 3,16, 4,21, 9,24,11, 26,15)$ |
| $ 9, 9, 9 $ | $1$ | $9$ | $( 1,18, 4,19, 8,24,12,25,15)( 2,16, 5,20, 9,22,10,26,13)( 3,17, 6,21, 7,23,11, 27,14)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1,19,12)( 2,20,10)( 3,21,11)( 4,24,15)( 5,22,13)( 6,23,14)( 7,27,17) ( 8,25,18)( 9,26,16)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1,20,11)( 2,21,12)( 3,19,10)( 4,22,14)( 5,23,15)( 6,24,13)( 7,25,16) ( 8,26,17)( 9,27,18)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1,21,10)( 2,19,11)( 3,20,12)( 4,23,13)( 5,24,14)( 6,22,15)( 7,26,18) ( 8,27,16)( 9,25,17)$ |
| $ 9, 9, 9 $ | $1$ | $9$ | $( 1,22,17,12, 5,27,19,13, 7)( 2,23,18,10, 6,25,20,14, 8)( 3,24,16,11, 4,26,21, 15, 9)$ |
| $ 9, 9, 9 $ | $1$ | $9$ | $( 1,23,16,12, 6,26,19,14, 9)( 2,24,17,10, 4,27,20,15, 7)( 3,22,18,11, 5,25,21, 13, 8)$ |
| $ 9, 9, 9 $ | $1$ | $9$ | $( 1,24,18,12, 4,25,19,15, 8)( 2,22,16,10, 5,26,20,13, 9)( 3,23,17,11, 6,27,21, 14, 7)$ |
| $ 9, 9, 9 $ | $1$ | $9$ | $( 1,25,24,19,18,15,12, 8, 4)( 2,26,22,20,16,13,10, 9, 5)( 3,27,23,21,17,14,11, 7, 6)$ |
| $ 9, 9, 9 $ | $1$ | $9$ | $( 1,26,23,19,16,14,12, 9, 6)( 2,27,24,20,17,15,10, 7, 4)( 3,25,22,21,18,13,11, 8, 5)$ |
| $ 9, 9, 9 $ | $1$ | $9$ | $( 1,27,22,19,17,13,12, 7, 5)( 2,25,23,20,18,14,10, 8, 6)( 3,26,24,21,16,15,11, 9, 4)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $27=3^{3}$ | magma: Order(G);
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| Cyclic: | no | magma: IsCyclic(G);
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| Abelian: | yes | magma: IsAbelian(G);
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| Solvable: | yes | magma: IsSolvable(G);
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| Nilpotency class: | $1$ | ||
| Label: | 27.2 | magma: IdentifyGroup(G);
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| Character table: not available. |
magma: CharacterTable(G);