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Magma
magma: G := TransitiveGroup(18, 20);
Group invariants
Abstract group: | $C_3^2:C_6$ | magma: IdentifyGroup(G);
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Order: | $54=2 \cdot 3^{3}$ | 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 | magma: NilpotencyClass(G);
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Group action invariants
Degree $n$: | $18$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $20$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Parity: | $-1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
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$\card{\Aut(F/K)}$: | $6$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | $(1,12,5,3,15,7)(2,11,6,4,16,8)(9,18,14,10,17,13)$, $(1,2)(3,17)(4,18)(5,10)(6,9)(7,8)(11,16)(12,15)(13,14)$ | magma: Generators(G);
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Low degree resolvents
$\card{(G/N)}$ Galois groups for stem field(s) $2$: $C_2$ $3$: $C_3$ $6$: $S_3$, $C_6$ $18$: $S_3\times C_3$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: $C_3$
Degree 6: $C_6$
Degree 9: $C_3^2 : S_3 $
Low degree siblings
9T11, 9T13, 18T21, 18T22, 27T11Siblings 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 | Index | Representative |
1A | $1^{18}$ | $1$ | $1$ | $0$ | $()$ |
2A | $2^{9}$ | $9$ | $2$ | $9$ | $( 1, 3)( 2, 4)( 5,10)( 6, 9)( 7, 8)(11,13)(12,14)(15,16)(17,18)$ |
3A | $3^{6}$ | $2$ | $3$ | $12$ | $( 1,17, 4)( 2,18, 3)( 5, 9, 8)( 6,10, 7)(11,15,14)(12,16,13)$ |
3B1 | $3^{6}$ | $3$ | $3$ | $12$ | $( 1,11, 9)( 2,12,10)( 3,13, 6)( 4,14, 5)( 7,18,16)( 8,17,15)$ |
3B-1 | $3^{6}$ | $3$ | $3$ | $12$ | $( 1, 9,11)( 2,10,12)( 3, 6,13)( 4, 5,14)( 7,16,18)( 8,15,17)$ |
3C | $3^{4},1^{6}$ | $6$ | $3$ | $8$ | $( 5, 9, 8)( 6,10, 7)(11,14,15)(12,13,16)$ |
3D1 | $3^{6}$ | $6$ | $3$ | $12$ | $( 1, 9,15)( 2,10,16)( 3, 6,12)( 4, 5,11)( 7,13,18)( 8,14,17)$ |
3D-1 | $3^{6}$ | $6$ | $3$ | $12$ | $( 1,11, 5)( 2,12, 6)( 3,13, 7)( 4,14, 8)( 9,17,15)(10,18,16)$ |
6A1 | $6^{3}$ | $9$ | $6$ | $15$ | $( 1, 6,11, 3, 9,13)( 2, 5,12, 4,10,14)( 7,15,18, 8,16,17)$ |
6A-1 | $6^{3}$ | $9$ | $6$ | $15$ | $( 1,13, 9, 3,11, 6)( 2,14,10, 4,12, 5)( 7,17,16, 8,18,15)$ |
Malle's constant $a(G)$: $1/8$
magma: ConjugacyClasses(G);
Character table
1A | 2A | 3A | 3B1 | 3B-1 | 3C | 3D1 | 3D-1 | 6A1 | 6A-1 | ||
Size | 1 | 9 | 2 | 3 | 3 | 6 | 6 | 6 | 9 | 9 | |
2 P | 1A | 1A | 3A | 3B-1 | 3B1 | 3C | 3D-1 | 3D1 | 3B1 | 3B-1 | |
3 P | 1A | 2A | 1A | 1A | 1A | 1A | 1A | 1A | 2A | 2A | |
Type | |||||||||||
54.5.1a | R | ||||||||||
54.5.1b | R | ||||||||||
54.5.1c1 | C | ||||||||||
54.5.1c2 | C | ||||||||||
54.5.1d1 | C | ||||||||||
54.5.1d2 | C | ||||||||||
54.5.2a | R | ||||||||||
54.5.2b1 | C | ||||||||||
54.5.2b2 | C | ||||||||||
54.5.6a | R |
magma: CharacterTable(G);
Regular extensions
Data not computed