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Magma
magma: G := TransitiveGroup(27, 17);
Group action invariants
| Degree $n$: | $27$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $17$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_3^2:C_9$ | ||
| Parity: | $1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $9$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,9,13,20,27,4,11,17,23)(2,7,14,21,25,5,12,18,24)(3,8,15,19,26,6,10,16,22), (4,22,14)(5,23,15)(6,24,13)(7,16,27)(8,17,25)(9,18,26) | 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, $C_3^2$ $27$: $C_9:C_3$ x 2, $C_3^2:C_3$, 27T2 Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $C_3$
Degree 9: $C_9$, $C_9:C_3$ x 2, $C_3^2:C_3$
Low degree siblings
27T17 x 2Siblings 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, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $3$ | $3$ | $( 4,14,22)( 5,15,23)( 6,13,24)( 7,27,16)( 8,25,17)( 9,26,18)$ |
| $ 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $3$ | $3$ | $( 4,22,14)( 5,23,15)( 6,24,13)( 7,16,27)( 8,17,25)( 9,18,26)$ |
| $ 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 $ | $3$ | $3$ | $( 1, 2, 3)( 4,15,24)( 5,13,22)( 6,14,23)( 7,25,18)( 8,26,16)( 9,27,17) (10,11,12)(19,20,21)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $3$ | $3$ | $( 1, 2, 3)( 4,23,13)( 5,24,14)( 6,22,15)( 7,17,26)( 8,18,27)( 9,16,25) (10,11,12)(19,20,21)$ |
| $ 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 $ | $3$ | $3$ | $( 1, 3, 2)( 4,13,23)( 5,14,24)( 6,15,22)( 7,26,17)( 8,27,18)( 9,25,16) (10,12,11)(19,21,20)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $3$ | $3$ | $( 1, 3, 2)( 4,24,15)( 5,22,13)( 6,23,14)( 7,18,25)( 8,16,26)( 9,17,27) (10,12,11)(19,21,20)$ |
| $ 9, 9, 9 $ | $3$ | $9$ | $( 1, 4, 9,11,13,17,20,23,27)( 2, 5, 7,12,14,18,21,24,25)( 3, 6, 8,10,15,16,19, 22,26)$ |
| $ 9, 9, 9 $ | $3$ | $9$ | $( 1, 4,18,11,13,25,20,23, 7)( 2, 5,16,12,14,26,21,24, 8)( 3, 6,17,10,15,27,19, 22, 9)$ |
| $ 9, 9, 9 $ | $3$ | $9$ | $( 1, 4,26,11,13, 8,20,23,16)( 2, 5,27,12,14, 9,21,24,17)( 3, 6,25,10,15, 7,19, 22,18)$ |
| $ 9, 9, 9 $ | $3$ | $9$ | $( 1, 5, 8,11,14,16,20,24,26)( 2, 6, 9,12,15,17,21,22,27)( 3, 4, 7,10,13,18,19, 23,25)$ |
| $ 9, 9, 9 $ | $3$ | $9$ | $( 1, 5,17,11,14,27,20,24, 9)( 2, 6,18,12,15,25,21,22, 7)( 3, 4,16,10,13,26,19, 23, 8)$ |
| $ 9, 9, 9 $ | $3$ | $9$ | $( 1, 5,25,11,14, 7,20,24,18)( 2, 6,26,12,15, 8,21,22,16)( 3, 4,27,10,13, 9,19, 23,17)$ |
| $ 9, 9, 9 $ | $3$ | $9$ | $( 1, 6, 7,11,15,18,20,22,25)( 2, 4, 8,12,13,16,21,23,26)( 3, 5, 9,10,14,17,19, 24,27)$ |
| $ 9, 9, 9 $ | $3$ | $9$ | $( 1, 6,16,11,15,26,20,22, 8)( 2, 4,17,12,13,27,21,23, 9)( 3, 5,18,10,14,25,19, 24, 7)$ |
| $ 9, 9, 9 $ | $3$ | $9$ | $( 1, 6,27,11,15, 9,20,22,17)( 2, 4,25,12,13, 7,21,23,18)( 3, 5,26,10,14, 8,19, 24,16)$ |
| $ 9, 9, 9 $ | $3$ | $9$ | $( 1, 7,23,20,25,13,11,18, 4)( 2, 8,24,21,26,14,12,16, 5)( 3, 9,22,19,27,15,10, 17, 6)$ |
| $ 9, 9, 9 $ | $3$ | $9$ | $( 1, 7,15,20,25, 6,11,18,22)( 2, 8,13,21,26, 4,12,16,23)( 3, 9,14,19,27, 5,10, 17,24)$ |
| $ 9, 9, 9 $ | $3$ | $9$ | $( 1, 7, 5,20,25,24,11,18,14)( 2, 8, 6,21,26,22,12,16,15)( 3, 9, 4,19,27,23,10, 17,13)$ |
| $ 9, 9, 9 $ | $3$ | $9$ | $( 1, 8,22,20,26,15,11,16, 6)( 2, 9,23,21,27,13,12,17, 4)( 3, 7,24,19,25,14,10, 18, 5)$ |
| $ 9, 9, 9 $ | $3$ | $9$ | $( 1, 8,14,20,26, 5,11,16,24)( 2, 9,15,21,27, 6,12,17,22)( 3, 7,13,19,25, 4,10, 18,23)$ |
| $ 9, 9, 9 $ | $3$ | $9$ | $( 1, 8, 4,20,26,23,11,16,13)( 2, 9, 5,21,27,24,12,17,14)( 3, 7, 6,19,25,22,10, 18,15)$ |
| $ 9, 9, 9 $ | $3$ | $9$ | $( 1, 9,24,20,27,14,11,17, 5)( 2, 7,22,21,25,15,12,18, 6)( 3, 8,23,19,26,13,10, 16, 4)$ |
| $ 9, 9, 9 $ | $3$ | $9$ | $( 1, 9,13,20,27, 4,11,17,23)( 2, 7,14,21,25, 5,12,18,24)( 3, 8,15,19,26, 6,10, 16,22)$ |
| $ 9, 9, 9 $ | $3$ | $9$ | $( 1, 9, 6,20,27,22,11,17,15)( 2, 7, 4,21,25,23,12,18,13)( 3, 8, 5,19,26,24,10, 16,14)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1,10,21)( 2,11,19)( 3,12,20)( 4,15,24)( 5,13,22)( 6,14,23)( 7,17,26) ( 8,18,27)( 9,16,25)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1,11,20)( 2,12,21)( 3,10,19)( 4,13,23)( 5,14,24)( 6,15,22)( 7,18,25) ( 8,16,26)( 9,17,27)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1,12,19)( 2,10,20)( 3,11,21)( 4,14,22)( 5,15,23)( 6,13,24)( 7,16,27) ( 8,17,25)( 9,18,26)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1,19,12)( 2,20,10)( 3,21,11)( 4,22,14)( 5,23,15)( 6,24,13)( 7,27,16) ( 8,25,17)( 9,26,18)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1,20,11)( 2,21,12)( 3,19,10)( 4,23,13)( 5,24,14)( 6,22,15)( 7,25,18) ( 8,26,16)( 9,27,17)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1,21,10)( 2,19,11)( 3,20,12)( 4,24,15)( 5,22,13)( 6,23,14)( 7,26,17) ( 8,27,18)( 9,25,16)$ |
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: | $2$ | ||
| Label: | 81.3 | magma: IdentifyGroup(G);
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| Character table: not available. |
magma: CharacterTable(G);