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Magma
magma: G := TransitiveGroup(18, 1);
Group action invariants
Degree $n$: | $18$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $1$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_{18}$ | ||
Parity: | $-1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $18$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,15,11,8,3,17,14,9,6,2,16,12,7,4,18,13,10,5) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $3$: $C_3$ $6$: $C_6$ $9$: $C_9$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: $C_3$
Degree 6: $C_6$
Degree 9: $C_9$
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 | Index | Representative |
1A | $1^{18}$ | $1$ | $1$ | $0$ | $()$ |
2A | $2^{9}$ | $1$ | $2$ | $9$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)$ |
3A1 | $3^{6}$ | $1$ | $3$ | $12$ | $( 1,14, 7)( 2,13, 8)( 3,16,10)( 4,15, 9)( 5,17,12)( 6,18,11)$ |
3A-1 | $3^{6}$ | $1$ | $3$ | $12$ | $( 1, 7,14)( 2, 8,13)( 3,10,16)( 4, 9,15)( 5,12,17)( 6,11,18)$ |
6A1 | $6^{3}$ | $1$ | $6$ | $15$ | $( 1, 8,14, 2, 7,13)( 3, 9,16, 4,10,15)( 5,11,17, 6,12,18)$ |
6A-1 | $6^{3}$ | $1$ | $6$ | $15$ | $( 1,13, 7, 2,14, 8)( 3,15,10, 4,16, 9)( 5,18,12, 6,17,11)$ |
9A1 | $9^{2}$ | $1$ | $9$ | $16$ | $( 1,10,18, 7,16, 6,14, 3,11)( 2, 9,17, 8,15, 5,13, 4,12)$ |
9A-1 | $9^{2}$ | $1$ | $9$ | $16$ | $( 1,18,16,14,11,10, 7, 6, 3)( 2,17,15,13,12, 9, 8, 5, 4)$ |
9A2 | $9^{2}$ | $1$ | $9$ | $16$ | $( 1,16,11, 7, 3,18,14,10, 6)( 2,15,12, 8, 4,17,13, 9, 5)$ |
9A-2 | $9^{2}$ | $1$ | $9$ | $16$ | $( 1, 3, 6, 7,10,11,14,16,18)( 2, 4, 5, 8, 9,12,13,15,17)$ |
9A4 | $9^{2}$ | $1$ | $9$ | $16$ | $( 1, 6,10,14,18, 3, 7,11,16)( 2, 5, 9,13,17, 4, 8,12,15)$ |
9A-4 | $9^{2}$ | $1$ | $9$ | $16$ | $( 1,11, 3,14, 6,16, 7,18,10)( 2,12, 4,13, 5,15, 8,17, 9)$ |
18A1 | $18$ | $1$ | $18$ | $17$ | $( 1, 9,18, 8,16, 5,14, 4,11, 2,10,17, 7,15, 6,13, 3,12)$ |
18A-1 | $18$ | $1$ | $18$ | $17$ | $( 1,12, 3,13, 6,15, 7,17,10, 2,11, 4,14, 5,16, 8,18, 9)$ |
18A5 | $18$ | $1$ | $18$ | $17$ | $( 1,17,16,13,11, 9, 7, 5, 3, 2,18,15,14,12,10, 8, 6, 4)$ |
18A-5 | $18$ | $1$ | $18$ | $17$ | $( 1, 4, 6, 8,10,12,14,15,18, 2, 3, 5, 7, 9,11,13,16,17)$ |
18A7 | $18$ | $1$ | $18$ | $17$ | $( 1,15,11, 8, 3,17,14, 9, 6, 2,16,12, 7, 4,18,13,10, 5)$ |
18A-7 | $18$ | $1$ | $18$ | $17$ | $( 1, 5,10,13,18, 4, 7,12,16, 2, 6, 9,14,17, 3, 8,11,15)$ |
magma: ConjugacyClasses(G);
Malle's constant $a(G)$: $1/9$
Group invariants
Order: | $18=2 \cdot 3^{2}$ | magma: Order(G);
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Cyclic: | yes | magma: IsCyclic(G);
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Abelian: | yes | magma: IsAbelian(G);
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Solvable: | yes | magma: IsSolvable(G);
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Nilpotency class: | $1$ | ||
Label: | 18.2 | magma: IdentifyGroup(G);
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Character table: |
1A | 2A | 3A1 | 3A-1 | 6A1 | 6A-1 | 9A1 | 9A-1 | 9A2 | 9A-2 | 9A4 | 9A-4 | 18A1 | 18A-1 | 18A5 | 18A-5 | 18A7 | 18A-7 | ||
Size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | |
2 P | 1A | 1A | 3A-1 | 3A1 | 3A1 | 3A-1 | 9A-2 | 9A-4 | 9A1 | 9A4 | 9A-1 | 9A2 | 9A-2 | 9A2 | 9A-4 | 9A4 | 9A1 | 9A-1 | |
3 P | 1A | 2A | 1A | 1A | 2A | 2A | 3A-1 | 3A1 | 3A-1 | 3A-1 | 3A1 | 3A1 | 6A1 | 6A-1 | 6A-1 | 6A1 | 6A1 | 6A-1 | |
Type | |||||||||||||||||||
18.2.1a | R | ||||||||||||||||||
18.2.1b | R | ||||||||||||||||||
18.2.1c1 | C | ||||||||||||||||||
18.2.1c2 | C | ||||||||||||||||||
18.2.1d1 | C | ||||||||||||||||||
18.2.1d2 | C | ||||||||||||||||||
18.2.1e1 | C | ||||||||||||||||||
18.2.1e2 | C | ||||||||||||||||||
18.2.1e3 | C | ||||||||||||||||||
18.2.1e4 | C | ||||||||||||||||||
18.2.1e5 | C | ||||||||||||||||||
18.2.1e6 | C | ||||||||||||||||||
18.2.1f1 | C | ||||||||||||||||||
18.2.1f2 | C | ||||||||||||||||||
18.2.1f3 | C | ||||||||||||||||||
18.2.1f4 | C | ||||||||||||||||||
18.2.1f5 | C | ||||||||||||||||||
18.2.1f6 | C |
magma: CharacterTable(G);