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Magma
magma: G := TransitiveGroup(18, 40);
Group action invariants
Degree $n$: | $18$ | 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: | $C_3:S_4$ | ||
Parity: | $-1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $2$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,18,2,17)(3,15,4,16)(5,14,6,13)(9,12)(10,11), (1,2)(3,6)(4,5)(7,17)(8,18)(9,15)(10,16)(11,13)(12,14), (1,5,3)(2,6,4)(7,11,10)(8,12,9)(13,17,16)(14,18,15) | 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 $18$: $C_3^2:C_2$ $24$: $S_4$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 3: $S_3$ x 4
Degree 6: $S_4$
Degree 9: $C_3^2:C_2$
Low degree siblings
12T44 x 3, 18T37, 24T79 x 3, 36T23, 36T56Siblings 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 $ | $1$ | $1$ | $()$ | |
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $3$ | $2$ | $( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)$ | |
$ 4, 4, 4, 2, 2, 1, 1 $ | $18$ | $4$ | $( 3, 5)( 4, 6)( 7,17, 8,18)( 9,15,10,16)(11,13,12,14)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $18$ | $2$ | $( 1, 2)( 3, 6)( 4, 5)( 7,17)( 8,18)( 9,15)(10,16)(11,13)(12,14)$ | |
$ 6, 6, 3, 3 $ | $6$ | $6$ | $( 1, 3, 5)( 2, 4, 6)( 7, 9,11, 8,10,12)(13,15,17,14,16,18)$ | |
$ 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 3, 5)( 2, 4, 6)( 7,10,11)( 8, 9,12)(13,16,17)(14,15,18)$ | |
$ 3, 3, 3, 3, 3, 3 $ | $8$ | $3$ | $( 1, 7,17)( 2, 8,18)( 3,10,13)( 4, 9,14)( 5,11,16)( 6,12,15)$ | |
$ 3, 3, 3, 3, 3, 3 $ | $8$ | $3$ | $( 1, 9,15)( 2,10,16)( 3,12,18)( 4,11,17)( 5, 8,14)( 6, 7,13)$ | |
$ 3, 3, 3, 3, 3, 3 $ | $8$ | $3$ | $( 1,11,14)( 2,12,13)( 3, 7,15)( 4, 8,16)( 5,10,18)( 6, 9,17)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $72=2^{3} \cdot 3^{2}$ | 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: | 72.43 | magma: IdentifyGroup(G);
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Character table: |
1A | 2A | 2B | 3A | 3B | 3C | 3D | 4A | 6A | ||
Size | 1 | 3 | 18 | 2 | 8 | 8 | 8 | 18 | 6 | |
2 P | 1A | 1A | 1A | 3A | 3B | 3C | 3D | 2A | 3A | |
3 P | 1A | 2A | 2B | 1A | 1A | 1A | 1A | 4A | 2A | |
Type | ||||||||||
72.43.1a | R | |||||||||
72.43.1b | R | |||||||||
72.43.2a | R | |||||||||
72.43.2b | R | |||||||||
72.43.2c | R | |||||||||
72.43.2d | R | |||||||||
72.43.3a | R | |||||||||
72.43.3b | R | |||||||||
72.43.6a | R |
magma: CharacterTable(G);