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Magma
magma: G := TransitiveGroup(18, 12);
Group invariants
Abstract group: | $C_6:S_3$ | magma: IdentifyGroup(G);
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Order: | $36=2^{2} \cdot 3^{2}$ | 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 | 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 $t$: | $12$ | 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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$\card{\Aut(F/K)}$: | $2$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | $(1,17)(2,18)(3,4)(5,13)(6,14)(7,12)(8,11)(9,16)(10,15)$, $(1,15)(2,16)(3,13)(4,14)(5,6)(7,9)(8,10)(11,17)(12,18)$, $(1,11)(2,12)(3,15)(4,16)(5,8)(6,7)(13,18)(14,17)$ | magma: Generators(G);
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Low degree resolvents
$\card{(G/N)}$ Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_2^2$ $6$: $S_3$ x 4 $12$: $D_{6}$ x 4 $18$: $C_3^2:C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: $S_3$ x 4
Degree 6: $D_{6}$ x 4
Degree 9: $C_3^2:C_2$
Low degree siblings
18T12, 36T8Siblings 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 | $2^{9}$ | $1$ | $2$ | $9$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)$ |
2B | $2^{8},1^{2}$ | $9$ | $2$ | $8$ | $( 1,11)( 2,12)( 3,15)( 4,16)( 5, 8)( 6, 7)(13,18)(14,17)$ |
2C | $2^{9}$ | $9$ | $2$ | $9$ | $( 1,12)( 2,11)( 3,16)( 4,15)( 5, 7)( 6, 8)( 9,10)(13,17)(14,18)$ |
3A | $3^{6}$ | $2$ | $3$ | $12$ | $( 1,13, 7)( 2,14, 8)( 3,15, 9)( 4,16,10)( 5,17,12)( 6,18,11)$ |
3B | $3^{6}$ | $2$ | $3$ | $12$ | $( 1,16, 6)( 2,15, 5)( 3,12, 8)( 4,11, 7)( 9,17,14)(10,18,13)$ |
3C | $3^{6}$ | $2$ | $3$ | $12$ | $( 1,11,10)( 2,12, 9)( 3,14, 5)( 4,13, 6)( 7,18,16)( 8,17,15)$ |
3D | $3^{6}$ | $2$ | $3$ | $12$ | $( 1, 4,18)( 2, 3,17)( 5, 8, 9)( 6, 7,10)(11,13,16)(12,14,15)$ |
6A | $6^{3}$ | $2$ | $6$ | $15$ | $( 1, 8,13, 2, 7,14)( 3,10,15, 4, 9,16)( 5,11,17, 6,12,18)$ |
6B | $6^{3}$ | $2$ | $6$ | $15$ | $( 1, 5,16, 2, 6,15)( 3, 7,12, 4, 8,11)( 9,13,17,10,14,18)$ |
6C | $6^{3}$ | $2$ | $6$ | $15$ | $( 1, 9,11, 2,10,12)( 3, 6,14, 4, 5,13)( 7,15,18, 8,16,17)$ |
6D | $6^{3}$ | $2$ | $6$ | $15$ | $( 1,17, 4, 2,18, 3)( 5,10, 8, 6, 9, 7)(11,15,13,12,16,14)$ |
Malle's constant $a(G)$: $1/8$
magma: ConjugacyClasses(G);
Character table
1A | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 6A | 6B | 6C | 6D | ||
Size | 1 | 1 | 9 | 9 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | |
2 P | 1A | 1A | 1A | 1A | 3A | 3B | 3C | 3D | 3A | 3B | 3C | 3D | |
3 P | 1A | 2A | 2B | 2C | 1A | 1A | 1A | 1A | 2A | 2A | 2A | 2A | |
Type | |||||||||||||
36.13.1a | R | ||||||||||||
36.13.1b | R | ||||||||||||
36.13.1c | R | ||||||||||||
36.13.1d | R | ||||||||||||
36.13.2a | R | ||||||||||||
36.13.2b | R | ||||||||||||
36.13.2c | R | ||||||||||||
36.13.2d | R | ||||||||||||
36.13.2e | R | ||||||||||||
36.13.2f | R | ||||||||||||
36.13.2g | R | ||||||||||||
36.13.2h | R |
magma: CharacterTable(G);
Regular extensions
Data not computed