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Magma
magma: G := TransitiveGroup(27, 36);
Group action invariants
Degree $n$: | $27$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $36$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_3\times S_3^2$ | ||
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: | (1,21,12)(2,19,10)(3,20,11)(4,17,15,27,22,7)(5,18,13,25,23,8)(6,16,14,26,24,9), (1,26,6)(2,27,4)(3,25,5)(7,23,10,18,15,20)(8,24,11,16,13,21)(9,22,12,17,14,19) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $3$: $C_3$ $4$: $C_2^2$ $6$: $S_3$ x 2, $C_6$ x 3 $12$: $D_{6}$ x 2, $C_6\times C_2$ $18$: $S_3\times C_3$ x 2 $36$: $S_3^2$, $C_6\times S_3$ x 2 Resolvents shown for degrees $\leq 47$
Subfields
Degree 9: $S_3\times C_3$ x 2, $S_3^2$
Low degree siblings
12T70, 18T43, 18T46 x 2, 36T80, 36T82 x 2, 36T92Siblings 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 $ | $1$ | $1$ | $()$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $3$ | $2$ | $( 7,18)( 8,16)( 9,17)(10,20)(11,21)(12,19)(13,24)(14,22)(15,23)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $3$ | $2$ | $( 4,27)( 5,25)( 6,26)( 7,15)( 8,13)( 9,14)(16,24)(17,22)(18,23)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1 $ | $9$ | $2$ | $( 4,27)( 5,25)( 6,26)( 7,23)( 8,24)( 9,22)(10,20)(11,21)(12,19)(13,16)(14,17) (15,18)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21) (22,23,24)(25,26,27)$ |
$ 6, 6, 6, 3, 3, 3 $ | $3$ | $6$ | $( 1, 2, 3)( 4, 5, 6)( 7,16, 9,18, 8,17)(10,21,12,20,11,19)(13,22,15,24,14,23) (25,26,27)$ |
$ 6, 6, 6, 3, 3, 3 $ | $3$ | $6$ | $( 1, 2, 3)( 4,25, 6,27, 5,26)( 7,13, 9,15, 8,14)(10,11,12)(16,22,18,24,17,23) (19,20,21)$ |
$ 6, 6, 6, 6, 3 $ | $9$ | $6$ | $( 1, 2, 3)( 4,25, 6,27, 5,26)( 7,24, 9,23, 8,22)(10,21,12,20,11,19) (13,17,15,16,14,18)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,12,11)(13,15,14)(16,18,17)(19,21,20) (22,24,23)(25,27,26)$ |
$ 6, 6, 6, 3, 3, 3 $ | $3$ | $6$ | $( 1, 3, 2)( 4, 6, 5)( 7,17, 8,18, 9,16)(10,19,11,20,12,21)(13,23,14,24,15,22) (25,27,26)$ |
$ 6, 6, 6, 3, 3, 3 $ | $3$ | $6$ | $( 1, 3, 2)( 4,26, 5,27, 6,25)( 7,14, 8,15, 9,13)(10,12,11)(16,23,17,24,18,22) (19,21,20)$ |
$ 6, 6, 6, 6, 3 $ | $9$ | $6$ | $( 1, 3, 2)( 4,26, 5,27, 6,25)( 7,22, 8,23, 9,24)(10,19,11,20,12,21) (13,18,14,16,15,17)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 4,25)( 2, 5,26)( 3, 6,27)( 7,11,14)( 8,12,15)( 9,10,13)(16,19,23) (17,20,24)(18,21,22)$ |
$ 6, 6, 6, 3, 3, 3 $ | $6$ | $6$ | $( 1, 4,25)( 2, 5,26)( 3, 6,27)( 7,21,14,18,11,22)( 8,19,15,16,12,23) ( 9,20,13,17,10,24)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 5,27)( 2, 6,25)( 3, 4,26)( 7,12,13)( 8,10,14)( 9,11,15)(16,20,22) (17,21,23)(18,19,24)$ |
$ 6, 6, 6, 3, 3, 3 $ | $6$ | $6$ | $( 1, 5,27)( 2, 6,25)( 3, 4,26)( 7,19,13,18,12,24)( 8,20,14,16,10,22) ( 9,21,15,17,11,23)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 6,26)( 2, 4,27)( 3, 5,25)( 7,10,15)( 8,11,13)( 9,12,14)(16,21,24) (17,19,22)(18,20,23)$ |
$ 6, 6, 6, 3, 3, 3 $ | $6$ | $6$ | $( 1, 6,26)( 2, 4,27)( 3, 5,25)( 7,20,15,18,10,23)( 8,21,13,16,11,24) ( 9,19,14,17,12,22)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $4$ | $3$ | $( 1, 7,23)( 2, 8,24)( 3, 9,22)( 4,11,16)( 5,12,17)( 6,10,18)(13,21,27) (14,19,25)(15,20,26)$ |
$ 6, 6, 6, 3, 3, 3 $ | $6$ | $6$ | $( 1, 7,20,26,10,18)( 2, 8,21,27,11,16)( 3, 9,19,25,12,17)( 4,13,24)( 5,14,22) ( 6,15,23)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $4$ | $3$ | $( 1, 8,22)( 2, 9,23)( 3, 7,24)( 4,12,18)( 5,10,16)( 6,11,17)(13,19,26) (14,20,27)(15,21,25)$ |
$ 6, 6, 6, 3, 3, 3 $ | $6$ | $6$ | $( 1, 8,19,26,11,17)( 2, 9,20,27,12,18)( 3, 7,21,25,10,16)( 4,14,23)( 5,15,24) ( 6,13,22)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $4$ | $3$ | $( 1, 9,24)( 2, 7,22)( 3, 8,23)( 4,10,17)( 5,11,18)( 6,12,16)(13,20,25) (14,21,26)(15,19,27)$ |
$ 6, 6, 6, 3, 3, 3 $ | $6$ | $6$ | $( 1, 9,21,26,12,16)( 2, 7,19,27,10,17)( 3, 8,20,25,11,18)( 4,15,22)( 5,13,23) ( 6,14,24)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1,10,20)( 2,11,21)( 3,12,19)( 4,13,24)( 5,14,22)( 6,15,23)( 7,18,26) ( 8,16,27)( 9,17,25)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1,11,19)( 2,12,20)( 3,10,21)( 4,14,23)( 5,15,24)( 6,13,22)( 7,16,25) ( 8,17,26)( 9,18,27)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1,12,21)( 2,10,19)( 3,11,20)( 4,15,22)( 5,13,23)( 6,14,24)( 7,17,27) ( 8,18,25)( 9,16,26)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $108=2^{2} \cdot 3^{3}$ | 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: | 108.38 | magma: IdentifyGroup(G);
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Character table: not available. |
magma: CharacterTable(G);