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Magma
magma: G := TransitiveGroup(27, 47);
Group action invariants
Degree $n$: | $27$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $47$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_3^2:C_{18}$ | ||
Parity: | $-1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $3$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (7,10,13)(8,11,14)(9,12,15)(16,23,21)(17,24,19)(18,22,20), (1,8,22,2,9,23,3,7,24)(4,15,16,25,10,19,6,14,18,27,12,21,5,13,17,26,11,20) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $3$: $C_3$ $6$: $S_3$, $C_6$ $9$: $C_9$ $18$: $S_3\times C_3$, $C_{18}$ $54$: $C_3^2 : C_6$, $C_9\times S_3$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $C_3$
Degree 9: $C_9$, $C_3^2 : S_3 $
Low degree siblings
18T82Siblings 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 $ | $6$ | $3$ | $( 7,10,13)( 8,11,14)( 9,12,15)(16,23,21)(17,24,19)(18,22,20)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $( 4,27)( 5,25)( 6,26)(10,13)(11,14)(12,15)(16,21)(17,19)(18,20)$ |
$ 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 $ | $6$ | $3$ | $( 1, 2, 3)( 4, 5, 6)( 7,11,15)( 8,12,13)( 9,10,14)(16,24,20)(17,22,21) (18,23,19)(25,26,27)$ |
$ 6, 6, 6, 3, 3, 3 $ | $9$ | $6$ | $( 1, 2, 3)( 4,25, 6,27, 5,26)( 7, 8, 9)(10,14,12,13,11,15)(16,19,18,21,17,20) (22,23,24)$ |
$ 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 $ | $6$ | $3$ | $( 1, 3, 2)( 4, 6, 5)( 7,12,14)( 8,10,15)( 9,11,13)(16,22,19)(17,23,20) (18,24,21)(25,27,26)$ |
$ 6, 6, 6, 3, 3, 3 $ | $9$ | $6$ | $( 1, 3, 2)( 4,26, 5,27, 6,25)( 7, 9, 8)(10,15,11,13,12,14)(16,20,17,21,18,19) (22,24,23)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 4,25)( 2, 5,26)( 3, 6,27)( 7,11,15)( 8,12,13)( 9,10,14)(16,19,22) (17,20,23)(18,21,24)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 5,27)( 2, 6,25)( 3, 4,26)( 7,12,14)( 8,10,15)( 9,11,13)(16,20,24) (17,21,22)(18,19,23)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 6,26)( 2, 4,27)( 3, 5,25)( 7,10,13)( 8,11,14)( 9,12,15)(16,21,23) (17,19,24)(18,20,22)$ |
$ 9, 9, 9 $ | $6$ | $9$ | $( 1, 7,16, 2, 8,17, 3, 9,18)( 4,11,19, 5,12,20, 6,10,21)(13,23,27,14,24,25,15, 22,26)$ |
$ 9, 9, 9 $ | $3$ | $9$ | $( 1, 7,23, 2, 8,24, 3, 9,22)( 4,11,17, 5,12,18, 6,10,16)(13,21,27,14,19,25,15, 20,26)$ |
$ 18, 9 $ | $9$ | $18$ | $( 1, 7,16, 4,14,19, 3, 9,18, 6,13,21, 2, 8,17, 5,15,20)(10,23,27,11,24,25,12, 22,26)$ |
$ 9, 9, 9 $ | $6$ | $9$ | $( 1, 8,18, 2, 9,16, 3, 7,17)( 4,12,21, 5,10,19, 6,11,20)(13,24,26,14,22,27,15, 23,25)$ |
$ 9, 9, 9 $ | $3$ | $9$ | $( 1, 8,22, 2, 9,23, 3, 7,24)( 4,12,16, 5,10,17, 6,11,18)(13,19,26,14,20,27,15, 21,25)$ |
$ 18, 9 $ | $9$ | $18$ | $( 1, 8,18, 4,15,21, 3, 7,17, 6,14,20, 2, 9,16, 5,13,19)(10,24,26,11,22,27,12, 23,25)$ |
$ 9, 9, 9 $ | $6$ | $9$ | $( 1, 9,17, 2, 7,18, 3, 8,16)( 4,10,20, 5,11,21, 6,12,19)(13,22,25,14,23,26,15, 24,27)$ |
$ 9, 9, 9 $ | $3$ | $9$ | $( 1, 9,24, 2, 7,22, 3, 8,23)( 4,10,18, 5,11,16, 6,12,17)(13,20,25,14,21,26,15, 19,27)$ |
$ 18, 9 $ | $9$ | $18$ | $( 1, 9,17, 4,13,20, 3, 8,16, 6,15,19, 2, 7,18, 5,14,21)(10,22,25,11,23,26,12, 24,27)$ |
$ 9, 9, 9 $ | $6$ | $9$ | $( 1,16, 8, 3,18, 7, 2,17, 9)( 4,19,12, 6,21,11, 5,20,10)(13,27,24,15,26,23,14, 25,22)$ |
$ 9, 9, 9 $ | $3$ | $9$ | $( 1,16,14, 3,18,13, 2,17,15)( 4,19, 9, 6,21, 8, 5,20, 7)(10,27,24,12,26,23,11, 25,22)$ |
$ 18, 9 $ | $9$ | $18$ | $( 1,16,11,25,20, 7, 2,17,12,26,21, 8, 3,18,10,27,19, 9)( 4,24,15, 6,23,14, 5, 22,13)$ |
$ 9, 9, 9 $ | $6$ | $9$ | $( 1,17, 7, 3,16, 9, 2,18, 8)( 4,20,11, 6,19,10, 5,21,12)(13,25,23,15,27,22,14, 26,24)$ |
$ 9, 9, 9 $ | $3$ | $9$ | $( 1,17,13, 3,16,15, 2,18,14)( 4,20, 8, 6,19, 7, 5,21, 9)(10,25,23,12,27,22,11, 26,24)$ |
$ 18, 9 $ | $9$ | $18$ | $( 1,17,10,25,21, 9, 2,18,11,26,19, 7, 3,16,12,27,20, 8)( 4,22,14, 6,24,13, 5, 23,15)$ |
$ 9, 9, 9 $ | $6$ | $9$ | $( 1,18, 9, 3,17, 8, 2,16, 7)( 4,21,10, 6,20,12, 5,19,11)(13,26,22,15,25,24,14, 27,23)$ |
$ 9, 9, 9 $ | $3$ | $9$ | $( 1,18,15, 3,17,14, 2,16,13)( 4,21, 7, 6,20, 9, 5,19, 8)(10,26,22,12,25,24,11, 27,23)$ |
$ 18, 9 $ | $9$ | $18$ | $( 1,18,12,25,19, 8, 2,16,10,26,20, 9, 3,17,11,27,21, 7)( 4,23,13, 6,22,15, 5, 24,14)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $162=2 \cdot 3^{4}$ | 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: | not nilpotent | ||
Label: | 162.4 | magma: IdentifyGroup(G);
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Character table: not available. |
magma: CharacterTable(G);