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Magma
magma: G := TransitiveGroup(27, 22);
Group action invariants
| Degree $n$: | $27$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $22$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_{27}:C_3$ | ||
| Parity: | $1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $9$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,11,21,6,15,22,7,18,25,3,10,20,5,14,24,9,17,27,2,12,19,4,13,23,8,16,26), (10,11,12)(13,14,15)(16,17,18)(19,21,20)(22,24,23)(25,27,26) | 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, $C_3^2$ $27$: 27T2 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, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $3$ | $3$ | $(10,11,12)(13,14,15)(16,17,18)(19,21,20)(22,24,23)(25,27,26)$ |
| $ 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $3$ | $3$ | $(10,12,11)(13,15,14)(16,18,17)(19,20,21)(22,23,24)(25,26,27)$ |
| $ 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 $ | $1$ | $3$ | $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,12,11)(13,15,14)(16,18,17)(19,21,20) (22,24,23)(25,27,26)$ |
| $ 9, 9, 9 $ | $3$ | $9$ | $( 1, 4, 9, 3, 6, 8, 2, 5, 7)(10,13,18,12,15,17,11,14,16)(19,23,26,21,22,25,20, 24,27)$ |
| $ 9, 9, 9 $ | $3$ | $9$ | $( 1, 4, 9, 3, 6, 8, 2, 5, 7)(10,14,17,12,13,16,11,15,18)(19,22,27,21,24,26,20, 23,25)$ |
| $ 9, 9, 9 $ | $1$ | $9$ | $( 1, 4, 9, 3, 6, 8, 2, 5, 7)(10,15,16,12,14,18,11,13,17)(19,24,25,21,23,27,20, 22,26)$ |
| $ 9, 9, 9 $ | $1$ | $9$ | $( 1, 5, 8, 3, 4, 7, 2, 6, 9)(10,13,18,12,15,17,11,14,16)(19,22,27,21,24,26,20, 23,25)$ |
| $ 9, 9, 9 $ | $1$ | $9$ | $( 1, 6, 7, 3, 5, 9, 2, 4, 8)(10,14,17,12,13,16,11,15,18)(19,23,26,21,22,25,20, 24,27)$ |
| $ 9, 9, 9 $ | $3$ | $9$ | $( 1, 7, 5, 2, 8, 6, 3, 9, 4)(10,16,14,11,17,15,12,18,13)(19,27,24,20,25,22,21, 26,23)$ |
| $ 9, 9, 9 $ | $1$ | $9$ | $( 1, 7, 5, 2, 8, 6, 3, 9, 4)(10,17,13,11,18,14,12,16,15)(19,26,22,20,27,23,21, 25,24)$ |
| $ 9, 9, 9 $ | $3$ | $9$ | $( 1, 7, 5, 2, 8, 6, 3, 9, 4)(10,18,15,11,16,13,12,17,14)(19,25,23,20,26,24,21, 27,22)$ |
| $ 9, 9, 9 $ | $1$ | $9$ | $( 1, 8, 4, 2, 9, 5, 3, 7, 6)(10,18,15,11,16,13,12,17,14)(19,27,24,20,25,22,21, 26,23)$ |
| $ 9, 9, 9 $ | $1$ | $9$ | $( 1, 9, 6, 2, 7, 4, 3, 8, 5)(10,16,14,11,17,15,12,18,13)(19,25,23,20,26,24,21, 27,22)$ |
| $ 27 $ | $3$ | $27$ | $( 1,10,19, 6,14,23, 7,17,26, 3,12,21, 5,13,22, 9,16,25, 2,11,20, 4,15,24, 8, 18,27)$ |
| $ 27 $ | $3$ | $27$ | $( 1,10,20, 6,14,24, 7,17,27, 3,12,19, 5,13,23, 9,16,26, 2,11,21, 4,15,22, 8, 18,25)$ |
| $ 27 $ | $3$ | $27$ | $( 1,10,21, 6,14,22, 7,17,25, 3,12,20, 5,13,24, 9,16,27, 2,11,19, 4,15,23, 8, 18,26)$ |
| $ 27 $ | $3$ | $27$ | $( 1,13,27, 5,18,21, 8,12,24, 3,15,26, 4,17,20, 7,11,23, 2,14,25, 6,16,19, 9, 10,22)$ |
| $ 27 $ | $3$ | $27$ | $( 1,13,25, 5,18,19, 8,12,22, 3,15,27, 4,17,21, 7,11,24, 2,14,26, 6,16,20, 9, 10,23)$ |
| $ 27 $ | $3$ | $27$ | $( 1,13,26, 5,18,20, 8,12,23, 3,15,25, 4,17,19, 7,11,22, 2,14,27, 6,16,21, 9, 10,24)$ |
| $ 27 $ | $3$ | $27$ | $( 1,16,23, 4,12,27, 9,14,20, 3,18,22, 6,11,26, 8,13,19, 2,17,24, 5,10,25, 7, 15,21)$ |
| $ 27 $ | $3$ | $27$ | $( 1,16,24, 4,12,25, 9,14,21, 3,18,23, 6,11,27, 8,13,20, 2,17,22, 5,10,26, 7, 15,19)$ |
| $ 27 $ | $3$ | $27$ | $( 1,16,22, 4,12,26, 9,14,19, 3,18,24, 6,11,25, 8,13,21, 2,17,23, 5,10,27, 7, 15,20)$ |
| $ 27 $ | $3$ | $27$ | $( 1,19,13, 7,26,11, 5,22,18, 2,20,14, 8,27,12, 6,23,16, 3,21,15, 9,25,10, 4, 24,17)$ |
| $ 27 $ | $3$ | $27$ | $( 1,19,15, 7,26,10, 5,22,17, 2,20,13, 8,27,11, 6,23,18, 3,21,14, 9,25,12, 4, 24,16)$ |
| $ 27 $ | $3$ | $27$ | $( 1,19,14, 7,26,12, 5,22,16, 2,20,15, 8,27,10, 6,23,17, 3,21,13, 9,25,11, 4, 24,18)$ |
| $ 27 $ | $3$ | $27$ | $( 1,22,12, 9,19,18, 6,25,13, 2,23,10, 7,20,16, 4,26,14, 3,24,11, 8,21,17, 5, 27,15)$ |
| $ 27 $ | $3$ | $27$ | $( 1,22,11, 9,19,17, 6,25,15, 2,23,12, 7,20,18, 4,26,13, 3,24,10, 8,21,16, 5, 27,14)$ |
| $ 27 $ | $3$ | $27$ | $( 1,22,10, 9,19,16, 6,25,14, 2,23,11, 7,20,17, 4,26,15, 3,24,12, 8,21,18, 5, 27,13)$ |
| $ 27 $ | $3$ | $27$ | $( 1,25,18, 8,22,15, 4,21,11, 2,26,16, 9,23,13, 5,19,12, 3,27,17, 7,24,14, 6, 20,10)$ |
| $ 27 $ | $3$ | $27$ | $( 1,25,17, 8,22,14, 4,21,10, 2,26,18, 9,23,15, 5,19,11, 3,27,16, 7,24,13, 6, 20,12)$ |
| $ 27 $ | $3$ | $27$ | $( 1,25,16, 8,22,13, 4,21,12, 2,26,17, 9,23,14, 5,19,10, 3,27,18, 7,24,15, 6, 20,11)$ |
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: | 81.6 | magma: IdentifyGroup(G);
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| Character table: not available. |
magma: CharacterTable(G);