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Magma
magma: G := TransitiveGroup(27, 108);
Group action invariants
Degree $n$: | $27$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $108$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_9^2:C_3$ | ||
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: | (10,13,18,11,14,16,12,15,17)(19,26,24,21,25,23,20,27,22), (1,22,17)(2,23,18)(3,24,16)(4,27,12)(5,25,10)(6,26,11)(7,20,14)(8,21,15)(9,19,13) | magma: Generators(G);
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Low degree resolvents
$\card{(G/N)}$ Galois groups for stem field(s) $3$: $C_3$ x 4 $9$: $C_3^2$ $27$: $C_3^2:C_3$ $81$: 27T23 Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $C_3$
Degree 9: $C_3^2:C_3$
Low degree siblings
27T108 x 2Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
Label | Cycle Type | Size | Order | Index | Representative |
1A | $1^{27}$ | $1$ | $1$ | $0$ | $()$ |
3A1 | $3^{9}$ | $1$ | $3$ | $18$ | $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24)(25,26,27)$ |
3A-1 | $3^{9}$ | $1$ | $3$ | $18$ | $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,12,11)(13,15,14)(16,18,17)(19,21,20)(22,24,23)(25,27,26)$ |
3B1 | $3^{6},1^{9}$ | $3$ | $3$ | $12$ | $(10,11,12)(13,14,15)(16,17,18)(19,21,20)(22,24,23)(25,27,26)$ |
3B-1 | $3^{6},1^{9}$ | $3$ | $3$ | $12$ | $(10,12,11)(13,15,14)(16,18,17)(19,20,21)(22,23,24)(25,26,27)$ |
3C1 | $3^{9}$ | $27$ | $3$ | $18$ | $( 1,22,17)( 2,23,18)( 3,24,16)( 4,27,12)( 5,25,10)( 6,26,11)( 7,20,14)( 8,21,15)( 9,19,13)$ |
3C-1 | $3^{9}$ | $27$ | $3$ | $18$ | $( 1,17,22)( 2,18,23)( 3,16,24)( 4,12,27)( 5,10,25)( 6,11,26)( 7,14,20)( 8,15,21)( 9,13,19)$ |
9A1 | $9^{3}$ | $3$ | $9$ | $24$ | $( 1, 9, 6, 3, 8, 5, 2, 7, 4)(10,18,14,12,17,13,11,16,15)(19,27,23,21,26,22,20,25,24)$ |
9A-1 | $9^{3}$ | $3$ | $9$ | $24$ | $( 1, 7, 5, 3, 9, 4, 2, 8, 6)(10,16,13,12,18,15,11,17,14)(19,25,22,21,27,24,20,26,23)$ |
9A2 | $9^{2},3^{3}$ | $3$ | $9$ | $22$ | $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,18,14,12,17,13,11,16,15)(19,23,26,20,24,27,21,22,25)$ |
9A-2 | $9^{2},3^{3}$ | $3$ | $9$ | $22$ | $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,18,14,12,17,13,11,16,15)(19,22,27,20,23,25,21,24,26)$ |
9A4 | $9^{3}$ | $3$ | $9$ | $24$ | $( 1, 5, 9, 2, 6, 7, 3, 4, 8)(10,13,18,11,14,16,12,15,17)(19,22,27,20,23,25,21,24,26)$ |
9A-4 | $9^{2},3^{3}$ | $3$ | $9$ | $22$ | $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,14,17,11,15,18,12,13,16)(19,26,24,21,25,23,20,27,22)$ |
9B1 | $9^{2},3^{3}$ | $3$ | $9$ | $22$ | $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,13,18,11,14,16,12,15,17)(19,25,22,21,27,24,20,26,23)$ |
9B-1 | $9^{2},3^{3}$ | $3$ | $9$ | $22$ | $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,17,15,12,16,14,11,18,13)(19,24,25,20,22,26,21,23,27)$ |
9B2 | $9^{2},3^{3}$ | $3$ | $9$ | $22$ | $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,16,13,12,18,15,11,17,14)(19,24,25,20,22,26,21,23,27)$ |
9B-2 | $9^{2},3^{3}$ | $3$ | $9$ | $22$ | $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,17,15,12,16,14,11,18,13)(19,23,26,20,24,27,21,22,25)$ |
9B4 | $9^{2},1^{9}$ | $3$ | $9$ | $16$ | $(10,13,18,11,14,16,12,15,17)(19,26,24,21,25,23,20,27,22)$ |
9B-4 | $9^{2},1^{9}$ | $3$ | $9$ | $16$ | $(10,14,17,11,15,18,12,13,16)(19,25,22,21,27,24,20,26,23)$ |
9C1 | $9^{2},3^{3}$ | $3$ | $9$ | $22$ | $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,15,16,11,13,17,12,14,18)(19,26,24,21,25,23,20,27,22)$ |
9C-1 | $9^{3}$ | $3$ | $9$ | $24$ | $( 1, 8, 4, 3, 7, 6, 2, 9, 5)(10,17,15,12,16,14,11,18,13)(19,26,24,21,25,23,20,27,22)$ |
9C2 | $9^{2},3^{3}$ | $3$ | $9$ | $22$ | $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,13,18,11,14,16,12,15,17)(19,27,23,21,26,22,20,25,24)$ |
9C-2 | $9^{3}$ | $3$ | $9$ | $24$ | $( 1, 4, 7, 2, 5, 8, 3, 6, 9)(10,15,16,11,13,17,12,14,18)(19,24,25,20,22,26,21,23,27)$ |
9C4 | $9^{2},1^{9}$ | $3$ | $9$ | $16$ | $(10,16,13,12,18,15,11,17,14)(19,23,26,20,24,27,21,22,25)$ |
9C-4 | $9^{2},3^{3}$ | $3$ | $9$ | $22$ | $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,14,17,11,15,18,12,13,16)(19,27,23,21,26,22,20,25,24)$ |
9D1 | $9^{2},1^{9}$ | $3$ | $9$ | $16$ | $(10,15,16,11,13,17,12,14,18)(19,27,23,21,26,22,20,25,24)$ |
9D-1 | $9^{2},1^{9}$ | $3$ | $9$ | $16$ | $(10,18,14,12,17,13,11,16,15)(19,24,25,20,22,26,21,23,27)$ |
9D2 | $9^{2},3^{3}$ | $3$ | $9$ | $22$ | $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,15,16,11,13,17,12,14,18)(19,25,22,21,27,24,20,26,23)$ |
9D-2 | $9^{2},1^{9}$ | $3$ | $9$ | $16$ | $(10,17,15,12,16,14,11,18,13)(19,22,27,20,23,25,21,24,26)$ |
9D4 | $9^{2},3^{3}$ | $3$ | $9$ | $22$ | $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,16,13,12,18,15,11,17,14)(19,22,27,20,23,25,21,24,26)$ |
9D-4 | $9^{3}$ | $3$ | $9$ | $24$ | $( 1, 6, 8, 2, 4, 9, 3, 5, 7)(10,14,17,11,15,18,12,13,16)(19,23,26,20,24,27,21,22,25)$ |
9E1 | $9^{3}$ | $27$ | $9$ | $24$ | $( 1,26,16, 3,25,18, 2,27,17)( 4,19,11, 6,21,10, 5,20,12)( 7,22,13, 9,24,15, 8,23,14)$ |
9E-1 | $9^{3}$ | $27$ | $9$ | $24$ | $( 1,10,23, 2,11,24, 3,12,22)( 4,15,25, 5,13,26, 6,14,27)( 7,16,21, 8,17,19, 9,18,20)$ |
9F1 | $9^{3}$ | $27$ | $9$ | $24$ | $( 1,19,18, 2,20,16, 3,21,17)( 4,22,10, 5,23,11, 6,24,12)( 7,27,15, 8,25,13, 9,26,14)$ |
9F-1 | $9^{3}$ | $27$ | $9$ | $24$ | $( 1,13,24, 3,15,23, 2,14,22)( 4,17,26, 6,16,25, 5,18,27)( 7,12,19, 9,11,21, 8,10,20)$ |
Malle's constant $a(G)$: $1/12$
magma: ConjugacyClasses(G);
Group invariants
Order: | $243=3^{5}$ | 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: | $4$ | ||
Label: | 243.25 | magma: IdentifyGroup(G);
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Character table: | 35 x 35 character table |
magma: CharacterTable(G);