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Magma
magma: G := TransitiveGroup(27, 43);
Group action invariants
Degree $n$: | $27$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $43$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $(C_3\times C_9).S_3$ | ||
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: | (1,21,17,2,19,18,3,20,16)(4,22,11,5,23,12,6,24,10)(7,26,13,8,27,14,9,25,15), (1,23,15,2,24,13,3,22,14)(4,26,18,5,27,16,6,25,17)(7,21,10,8,19,11,9,20,12), (1,27,2,26,3,25)(4,24,5,23,6,22)(7,19,8,21,9,20)(10,12)(13,16)(14,18)(15,17) | 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
There are no siblings with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
Label | 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 $ | $3$ | $3$ | $(10,12,11)(13,15,14)(16,18,17)(19,20,21)(22,23,24)(25,26,27)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ | $27$ | $2$ | $( 2, 3)( 4, 8)( 5, 7)( 6, 9)(10,25)(11,27)(12,26)(13,23)(14,22)(15,24)(16,19) (17,21)(18,20)$ | |
$ 6, 6, 6, 2, 2, 2, 2, 1 $ | $27$ | $6$ | $( 2, 3)( 4, 8)( 5, 7)( 6, 9)(10,26,11,25,12,27)(13,24,14,23,15,22) (16,20,17,19,18,21)$ | |
$ 6, 6, 6, 2, 2, 2, 2, 1 $ | $27$ | $6$ | $( 2, 3)( 4, 8)( 5, 7)( 6, 9)(10,27,12,25,11,26)(13,22,15,23,14,24) (16,21,18,19,17,20)$ | |
$ 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 $ | $6$ | $9$ | $( 1, 4, 9, 2, 5, 7, 3, 6, 8)(10,13,18,11,14,16,12,15,17)(19,23,26,20,24,27,21, 22,25)$ | |
$ 9, 9, 9 $ | $6$ | $9$ | $( 1, 4, 9, 2, 5, 7, 3, 6, 8)(10,15,16,11,13,17,12,14,18)(19,24,25,20,22,26,21, 23,27)$ | |
$ 9, 9, 9 $ | $6$ | $9$ | $( 1, 5, 8, 2, 6, 9, 3, 4, 7)(10,13,18,11,14,16,12,15,17)(19,22,27,20,23,25,21, 24,26)$ | |
$ 9, 9, 9 $ | $18$ | $9$ | $( 1,10,25, 3,12,27, 2,11,26)( 4,14,19, 6,13,21, 5,15,20)( 7,18,24, 9,17,23, 8, 16,22)$ | |
$ 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 $ | $18$ | $9$ | $( 1,16,21, 3,18,20, 2,17,19)( 4,10,22, 6,12,24, 5,11,23)( 7,15,26, 9,14,25, 8, 13,27)$ |
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: | 162.22 | magma: IdentifyGroup(G);
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Character table: |
1A | 2A | 3A | 3B1 | 3B-1 | 6A1 | 6A-1 | 9A1 | 9A2 | 9A4 | 9B | 9C | 9D | ||
Size | 1 | 27 | 2 | 3 | 3 | 27 | 27 | 6 | 6 | 6 | 18 | 18 | 18 | |
2 P | 1A | 1A | 3A | 3B-1 | 3B1 | 3B1 | 3B-1 | 9A2 | 9A4 | 9A1 | 9B | 9C | 9D | |
3 P | 1A | 2A | 1A | 1A | 1A | 2A | 2A | 3A | 3A | 3A | 3A | 3A | 3A | |
Type | ||||||||||||||
162.22.1a | R | |||||||||||||
162.22.1b | R | |||||||||||||
162.22.2a | R | |||||||||||||
162.22.2b | R | |||||||||||||
162.22.2c | R | |||||||||||||
162.22.2d | R | |||||||||||||
162.22.3a1 | C | |||||||||||||
162.22.3a2 | C | |||||||||||||
162.22.3b1 | C | |||||||||||||
162.22.3b2 | C | |||||||||||||
162.22.6a1 | R | |||||||||||||
162.22.6a2 | R | |||||||||||||
162.22.6a3 | R |
magma: CharacterTable(G);