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Magma
magma: G := TransitiveGroup(27, 40);
Group action invariants
Degree $n$: | $27$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $40$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $\He_3.C_6$ | ||
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: | (1,23,15)(2,24,13)(3,22,14)(4,26,17)(5,27,18)(6,25,16)(7,21,12)(8,19,10)(9,20,11), (1,25)(2,26)(3,27)(4,21)(5,19)(6,20)(7,22)(8,23)(9,24) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $3$: $C_3$ $6$: $S_3$, $C_6$ $18$: $S_3\times C_3$ $54$: $C_3^2 : C_6$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 9: $S_3\times C_3$
Low degree siblings
27T49Siblings 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 | 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 $ | $6$ | $3$ | $(10,11,12)(13,14,15)(16,17,18)(19,21,20)(22,24,23)(25,27,26)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $(10,25)(11,26)(12,27)(13,21)(14,19)(15,20)(16,24)(17,22)(18,23)$ | |
$ 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 $ | $9$ | $6$ | $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,25,12,27,11,26)(13,21,15,20,14,19) (16,24,18,23,17,22)$ | |
$ 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 $ | $9$ | $6$ | $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,25,11,26,12,27)(13,21,14,19,15,20) (16,24,17,22,18,23)$ | |
$ 9, 9, 9 $ | $3$ | $9$ | $( 1, 4, 8, 2, 5, 9, 3, 6, 7)(10,13,18,11,14,16,12,15,17)(19,23,25,20,24,26,21, 22,27)$ | |
$ 9, 9, 9 $ | $3$ | $9$ | $( 1, 4, 8, 2, 5, 9, 3, 6, 7)(10,14,17,11,15,18,12,13,16)(19,22,26,20,23,27,21, 24,25)$ | |
$ 18, 9 $ | $9$ | $18$ | $( 1, 4, 8, 2, 5, 9, 3, 6, 7)(10,19,17,26,15,23,12,21,16,25,14,22,11,20,18,27, 13,24)$ | |
$ 9, 9, 9 $ | $3$ | $9$ | $( 1, 5, 7, 2, 6, 8, 3, 4, 9)(10,15,16,11,13,17,12,14,18)(19,23,25,20,24,26,21, 22,27)$ | |
$ 18, 9 $ | $9$ | $18$ | $( 1, 5, 7, 2, 6, 8, 3, 4, 9)(10,19,16,25,13,24,12,21,18,27,15,23,11,20,17,26, 14,22)$ | |
$ 18, 9 $ | $9$ | $18$ | $( 1, 6, 9, 2, 4, 7, 3, 5, 8)(10,19,18,27,14,22,12,21,17,26,13,24,11,20,16,25, 15,23)$ | |
$ 9, 9, 9 $ | $3$ | $9$ | $( 1, 7, 6, 3, 9, 5, 2, 8, 4)(10,16,13,12,18,15,11,17,14)(19,25,24,21,27,23,20, 26,22)$ | |
$ 9, 9, 9 $ | $3$ | $9$ | $( 1, 7, 6, 3, 9, 5, 2, 8, 4)(10,17,15,12,16,14,11,18,13)(19,27,22,21,26,24,20, 25,23)$ | |
$ 18, 9 $ | $9$ | $18$ | $( 1, 7, 6, 3, 9, 5, 2, 8, 4)(10,22,13,25,18,21,11,23,14,26,16,19,12,24,15,27, 17,20)$ | |
$ 9, 9, 9 $ | $3$ | $9$ | $( 1, 8, 5, 3, 7, 4, 2, 9, 6)(10,16,13,12,18,15,11,17,14)(19,27,22,21,26,24,20, 25,23)$ | |
$ 18, 9 $ | $9$ | $18$ | $( 1, 8, 5, 3, 7, 4, 2, 9, 6)(10,22,15,27,16,19,11,23,13,25,17,20,12,24,14,26, 18,21)$ | |
$ 18, 9 $ | $9$ | $18$ | $( 1, 9, 4, 3, 8, 6, 2, 7, 5)(10,22,14,26,17,20,11,23,15,27,18,21,12,24,13,25, 16,19)$ | |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $18$ | $3$ | $( 1,10,25)( 2,11,26)( 3,12,27)( 4,13,21)( 5,14,19)( 6,15,20)( 7,17,22) ( 8,18,23)( 9,16,24)$ | |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $18$ | $3$ | $( 1,13,24)( 2,14,22)( 3,15,23)( 4,18,27)( 5,16,25)( 6,17,26)( 7,10,19) ( 8,11,20)( 9,12,21)$ | |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $18$ | $3$ | $( 1,16,19)( 2,17,20)( 3,18,21)( 4,12,24)( 5,10,22)( 6,11,23)( 7,14,26) ( 8,15,27)( 9,13,25)$ |
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: | 162.14 | magma: IdentifyGroup(G);
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Character table: |
1A | 2A | 3A1 | 3A-1 | 3B | 3C | 3D1 | 3D-1 | 6A1 | 6A-1 | 9A1 | 9A-1 | 9A2 | 9A-2 | 9A4 | 9A-4 | 18A1 | 18A-1 | 18A5 | 18A-5 | 18A7 | 18A-7 | ||
Size | 1 | 9 | 1 | 1 | 6 | 18 | 18 | 18 | 9 | 9 | 3 | 3 | 3 | 3 | 3 | 3 | 9 | 9 | 9 | 9 | 9 | 9 | |
2 P | 1A | 1A | 3A-1 | 3A1 | 3B | 3C | 3D-1 | 3D1 | 3A1 | 3A-1 | 9A2 | 9A-2 | 9A4 | 9A-4 | 9A-1 | 9A1 | 9A-1 | 9A4 | 9A2 | 9A-4 | 9A-2 | 9A1 | |
3 P | 1A | 2A | 1A | 1A | 1A | 1A | 1A | 1A | 2A | 2A | 3A1 | 3A-1 | 3A-1 | 3A1 | 3A1 | 3A-1 | 6A-1 | 6A1 | 6A-1 | 6A-1 | 6A1 | 6A1 | |
Type | |||||||||||||||||||||||
162.14.1a | R | ||||||||||||||||||||||
162.14.1b | R | ||||||||||||||||||||||
162.14.1c1 | C | ||||||||||||||||||||||
162.14.1c2 | C | ||||||||||||||||||||||
162.14.1d1 | C | ||||||||||||||||||||||
162.14.1d2 | C | ||||||||||||||||||||||
162.14.2a | R | ||||||||||||||||||||||
162.14.2b1 | C | ||||||||||||||||||||||
162.14.2b2 | C | ||||||||||||||||||||||
162.14.3a1 | C | ||||||||||||||||||||||
162.14.3a2 | C | ||||||||||||||||||||||
162.14.3a3 | C | ||||||||||||||||||||||
162.14.3a4 | C | ||||||||||||||||||||||
162.14.3a5 | C | ||||||||||||||||||||||
162.14.3a6 | C | ||||||||||||||||||||||
162.14.3b1 | C | ||||||||||||||||||||||
162.14.3b2 | C | ||||||||||||||||||||||
162.14.3b3 | C | ||||||||||||||||||||||
162.14.3b4 | C | ||||||||||||||||||||||
162.14.3b5 | C | ||||||||||||||||||||||
162.14.3b6 | C | ||||||||||||||||||||||
162.14.6a | R |
magma: CharacterTable(G);