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Magma
magma: G := TransitiveGroup(27, 42);
Group action invariants
Degree $n$: | $27$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $42$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $\He_3.S_3$ | ||
Parity: | $-1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $1$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,12,3,10,2,11)(4,18,6,16,5,17)(7,14,9,15,8,13)(19,23)(20,22)(21,24)(25,26), (1,26,3,27,2,25)(4,23,6,24,5,22)(7,21,9,19,8,20)(11,12)(13,17)(14,16)(15,18), (1,16,19,2,17,20,3,18,21)(4,10,23,5,11,24,6,12,22)(7,15,27,8,13,25,9,14,26) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $6$: $S_3$ x 4 $18$: $C_3^2:C_2$ $54$: $(C_3^2:C_3):C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $S_3$ x 4
Degree 9: $C_3^2:C_2$
Low degree siblings
27T68Siblings 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 $ | $3$ | $3$ | $(10,11,12)(13,14,15)(16,17,18)(19,21,20)(22,24,23)(25,27,26)$ |
$ 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $3$ | $3$ | $(10,12,11)(13,15,14)(16,18,17)(19,20,21)(22,23,24)(25,26,27)$ |
$ 6, 6, 6, 2, 2, 2, 2, 1 $ | $27$ | $6$ | $( 2, 3)( 4, 8)( 5, 7)( 6, 9)(10,25,11,27,12,26)(13,23,14,22,15,24) (16,19,17,21,18,20)$ |
$ 6, 6, 6, 2, 2, 2, 2, 1 $ | $27$ | $6$ | $( 2, 3)( 4, 8)( 5, 7)( 6, 9)(10,26,12,27,11,25)(13,24,15,22,14,23) (16,20,18,21,17,19)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ | $27$ | $2$ | $( 2, 3)( 4, 8)( 5, 7)( 6, 9)(10,27)(11,26)(12,25)(13,22)(14,24)(15,23)(16,21) (17,20)(18,19)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21) (22,23,24)(25,26,27)$ |
$ 9, 9, 9 $ | $6$ | $9$ | $( 1, 4, 9, 2, 5, 7, 3, 6, 8)(10,13,18,11,14,16,12,15,17)(19,22,27,20,23,25,21, 24,26)$ |
$ 9, 9, 9 $ | $6$ | $9$ | $( 1, 4, 9, 2, 5, 7, 3, 6, 8)(10,14,17,11,15,18,12,13,16)(19,24,25,20,22,26,21, 23,27)$ |
$ 9, 9, 9 $ | $6$ | $9$ | $( 1, 5, 8, 2, 6, 9, 3, 4, 7)(10,13,18,11,14,16,12,15,17)(19,24,25,20,22,26,21, 23,27)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $18$ | $3$ | $( 1,10,25)( 2,11,26)( 3,12,27)( 4,14,19)( 5,15,20)( 6,13,21)( 7,18,24) ( 8,16,22)( 9,17,23)$ |
$ 9, 9, 9 $ | $18$ | $9$ | $( 1,13,22, 3,15,24, 2,14,23)( 4,16,25, 6,18,27, 5,17,26)( 7,11,20, 9,10,19, 8, 12,21)$ |
$ 9, 9, 9 $ | $18$ | $9$ | $( 1,16,21, 2,17,19, 3,18,20)( 4,10,22, 5,11,23, 6,12,24)( 7,15,26, 8,13,27, 9, 14,25)$ |
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: | 162.20 | magma: IdentifyGroup(G);
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Character table: |
2 1 1 1 1 1 1 . . . . . . . 3 4 3 3 1 1 1 4 3 3 3 2 2 2 1a 3a 3b 6a 6b 2a 3c 9a 9b 9c 3d 9d 9e 2P 1a 3b 3a 3a 3b 1a 3c 9c 9a 9b 3d 9d 9e 3P 1a 1a 1a 2a 2a 2a 1a 3c 3c 3c 1a 3c 3c 5P 1a 3b 3a 6b 6a 2a 3c 9b 9c 9a 3d 9d 9e 7P 1a 3a 3b 6a 6b 2a 3c 9c 9a 9b 3d 9d 9e X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 1 1 -1 -1 -1 1 1 1 1 1 1 1 X.3 2 2 2 . . . 2 2 2 2 -1 -1 -1 X.4 2 2 2 . . . 2 -1 -1 -1 2 -1 -1 X.5 2 2 2 . . . 2 -1 -1 -1 -1 -1 2 X.6 2 2 2 . . . 2 -1 -1 -1 -1 2 -1 X.7 3 A /A B /B -1 3 . . . . . . X.8 3 /A A /B B -1 3 . . . . . . X.9 3 A /A -B -/B 1 3 . . . . . . X.10 3 /A A -/B -B 1 3 . . . . . . X.11 6 . . . . . -3 C E D . . . X.12 6 . . . . . -3 D C E . . . X.13 6 . . . . . -3 E D C . . . A = 3*E(3)^2 = (-3-3*Sqrt(-3))/2 = -3-3b3 B = -E(3) = (1-Sqrt(-3))/2 = -b3 C = -2*E(9)^2-E(9)^4-E(9)^5-2*E(9)^7 D = E(9)^2-E(9)^4-E(9)^5+E(9)^7 E = E(9)^2+2*E(9)^4+2*E(9)^5+E(9)^7 |
magma: CharacterTable(G);