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Magma
magma: G := TransitiveGroup(27, 1);
Group action invariants
| Degree $n$: | $27$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $1$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_{27}$ | ||
| 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: | (1,6,9,12,14,16,19,23,27,2,4,7,10,15,17,20,24,25,3,5,8,11,13,18,21,22,26) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $3$: $C_3$ $9$: $C_9$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $C_3$
Degree 9: $C_9$
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 $ | $1$ | $3$ | $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,12,11)(13,15,14)(16,18,17)(19,21,20) (22,24,23)(25,27,26)$ |
| $ 27 $ | $1$ | $27$ | $( 1, 4, 8,12,15,18,19,24,26, 2, 5, 9,10,13,16,20,22,27, 3, 6, 7,11,14,17,21, 23,25)$ |
| $ 27 $ | $1$ | $27$ | $( 1, 5, 7,12,13,17,19,22,25, 2, 6, 8,10,14,18,20,23,26, 3, 4, 9,11,15,16,21, 24,27)$ |
| $ 27 $ | $1$ | $27$ | $( 1, 6, 9,12,14,16,19,23,27, 2, 4, 7,10,15,17,20,24,25, 3, 5, 8,11,13,18,21, 22,26)$ |
| $ 27 $ | $1$ | $27$ | $( 1, 7,13,19,25, 6,10,18,23, 3, 9,15,21,27, 5,12,17,22, 2, 8,14,20,26, 4,11, 16,24)$ |
| $ 27 $ | $1$ | $27$ | $( 1, 8,15,19,26, 5,10,16,22, 3, 7,14,21,25, 4,12,18,24, 2, 9,13,20,27, 6,11, 17,23)$ |
| $ 27 $ | $1$ | $27$ | $( 1, 9,14,19,27, 4,10,17,24, 3, 8,13,21,26, 6,12,16,23, 2, 7,15,20,25, 5,11, 18,22)$ |
| $ 9, 9, 9 $ | $1$ | $9$ | $( 1,10,21, 2,11,19, 3,12,20)( 4,13,23, 5,14,24, 6,15,22)( 7,18,27, 8,16,25, 9, 17,26)$ |
| $ 9, 9, 9 $ | $1$ | $9$ | $( 1,11,20, 2,12,21, 3,10,19)( 4,14,22, 5,15,23, 6,13,24)( 7,16,26, 8,17,27, 9, 18,25)$ |
| $ 9, 9, 9 $ | $1$ | $9$ | $( 1,12,19, 2,10,20, 3,11,21)( 4,15,24, 5,13,22, 6,14,23)( 7,17,25, 8,18,26, 9, 16,27)$ |
| $ 27 $ | $1$ | $27$ | $( 1,13,25,10,23, 9,21, 5,17, 2,14,26,11,24, 7,19, 6,18, 3,15,27,12,22, 8,20, 4,16)$ |
| $ 27 $ | $1$ | $27$ | $( 1,14,27,10,24, 8,21, 6,16, 2,15,25,11,22, 9,19, 4,17, 3,13,26,12,23, 7,20, 5,18)$ |
| $ 27 $ | $1$ | $27$ | $( 1,15,26,10,22, 7,21, 4,18, 2,13,27,11,23, 8,19, 5,16, 3,14,25,12,24, 9,20, 6,17)$ |
| $ 27 $ | $1$ | $27$ | $( 1,16, 4,20, 8,22,12,27,15, 3,18, 6,19, 7,24,11,26,14, 2,17, 5,21, 9,23,10, 25,13)$ |
| $ 27 $ | $1$ | $27$ | $( 1,17, 6,20, 9,24,12,25,14, 3,16, 5,19, 8,23,11,27,13, 2,18, 4,21, 7,22,10, 26,15)$ |
| $ 27 $ | $1$ | $27$ | $( 1,18, 5,20, 7,23,12,26,13, 3,17, 4,19, 9,22,11,25,15, 2,16, 6,21, 8,24,10, 27,14)$ |
| $ 9, 9, 9 $ | $1$ | $9$ | $( 1,19,10, 3,21,12, 2,20,11)( 4,24,13, 6,23,15, 5,22,14)( 7,25,18, 9,27,17, 8, 26,16)$ |
| $ 9, 9, 9 $ | $1$ | $9$ | $( 1,20,12, 3,19,11, 2,21,10)( 4,22,15, 6,24,14, 5,23,13)( 7,26,17, 9,25,16, 8, 27,18)$ |
| $ 9, 9, 9 $ | $1$ | $9$ | $( 1,21,11, 3,20,10, 2,19,12)( 4,23,14, 6,22,13, 5,24,15)( 7,27,16, 9,26,18, 8, 25,17)$ |
| $ 27 $ | $1$ | $27$ | $( 1,22,18,11, 5,25,20,15, 7, 2,23,16,12, 6,26,21,13, 8, 3,24,17,10, 4,27,19, 14, 9)$ |
| $ 27 $ | $1$ | $27$ | $( 1,23,17,11, 6,27,20,13, 9, 2,24,18,12, 4,25,21,14, 7, 3,22,16,10, 5,26,19, 15, 8)$ |
| $ 27 $ | $1$ | $27$ | $( 1,24,16,11, 4,26,20,14, 8, 2,22,17,12, 5,27,21,15, 9, 3,23,18,10, 6,25,19, 13, 7)$ |
| $ 27 $ | $1$ | $27$ | $( 1,25,23,21,17,14,11, 7, 6, 3,27,22,20,16,13,10, 9, 5, 2,26,24,19,18,15,12, 8, 4)$ |
| $ 27 $ | $1$ | $27$ | $( 1,26,22,21,18,13,11, 8, 5, 3,25,24,20,17,15,10, 7, 4, 2,27,23,19,16,14,12, 9, 6)$ |
| $ 27 $ | $1$ | $27$ | $( 1,27,24,21,16,15,11, 9, 4, 3,26,23,20,18,14,10, 8, 6, 2,25,22,19,17,13,12, 7, 5)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $27=3^{3}$ | magma: Order(G);
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| Cyclic: | yes | 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: | 27.1 | magma: IdentifyGroup(G);
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| Character table: not available. |
magma: CharacterTable(G);