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Magma
magma: G := TransitiveGroup(18, 6);
Group invariants
Abstract group: | $S_3 \times C_6$ | 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 $t$: | $6$ | 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: | $(1,15,8,4,13,9)(2,16,7,3,14,10)(5,17,12)(6,18,11)$, $(1,3,18,2,4,17)(5,8,10,6,7,9)(11,14,15,12,13,16)$ | magma: Generators(G);
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Low degree resolvents
$\card{(G/N)}$ Galois groups for stem field(s) $2$: $C_2$ x 3 $3$: $C_3$ $4$: $C_2^2$ $6$: $S_3$, $C_6$ x 3 $12$: $D_{6}$, $C_6\times C_2$ $18$: $S_3\times C_3$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 9: $S_3\times C_3$
Low degree siblings
12T18, 18T6, 36T6Siblings 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 | $2^{9}$ | $3$ | $2$ | $9$ | $( 1, 3)( 2, 4)( 5, 6)( 7, 9)( 8,10)(11,12)(13,16)(14,15)(17,18)$ |
2C | $2^{6},1^{6}$ | $3$ | $2$ | $6$ | $( 3,17)( 4,18)( 5,10)( 6, 9)(11,15)(12,16)$ |
3A1 | $3^{6}$ | $1$ | $3$ | $12$ | $( 1, 8,13)( 2, 7,14)( 3,10,16)( 4, 9,15)( 5,12,17)( 6,11,18)$ |
3A-1 | $3^{6}$ | $1$ | $3$ | $12$ | $( 1,13, 8)( 2,14, 7)( 3,16,10)( 4,15, 9)( 5,17,12)( 6,18,11)$ |
3B | $3^{6}$ | $2$ | $3$ | $12$ | $( 1, 4,18)( 2, 3,17)( 5, 7,10)( 6, 8, 9)(11,13,15)(12,14,16)$ |
3C1 | $3^{6}$ | $2$ | $3$ | $12$ | $( 1,15, 6)( 2,16, 5)( 3,12, 7)( 4,11, 8)( 9,18,13)(10,17,14)$ |
3C-1 | $3^{6}$ | $2$ | $3$ | $12$ | $( 1, 9,11)( 2,10,12)( 3, 5,14)( 4, 6,13)( 7,16,17)( 8,15,18)$ |
6A1 | $6^{3}$ | $1$ | $6$ | $15$ | $( 1, 7,13, 2, 8,14)( 3, 9,16, 4,10,15)( 5,11,17, 6,12,18)$ |
6A-1 | $6^{3}$ | $1$ | $6$ | $15$ | $( 1,14, 8, 2,13, 7)( 3,15,10, 4,16, 9)( 5,18,12, 6,17,11)$ |
6B | $6^{3}$ | $2$ | $6$ | $15$ | $( 1,12, 9, 2,11,10)( 3,13, 5, 4,14, 6)( 7,18,16, 8,17,15)$ |
6C1 | $6^{3}$ | $2$ | $6$ | $15$ | $( 1,17, 4, 2,18, 3)( 5, 9, 7, 6,10, 8)(11,16,13,12,15,14)$ |
6C-1 | $6^{3}$ | $2$ | $6$ | $15$ | $( 1, 5,15, 2, 6,16)( 3, 8,12, 4, 7,11)( 9,14,18,10,13,17)$ |
6D1 | $6^{3}$ | $3$ | $6$ | $15$ | $( 1,16, 8, 3,13,10)( 2,15, 7, 4,14, 9)( 5,18,12, 6,17,11)$ |
6D-1 | $6^{3}$ | $3$ | $6$ | $15$ | $( 1,10,13, 3, 8,16)( 2, 9,14, 4, 7,15)( 5,11,17, 6,12,18)$ |
6E1 | $6^{2},3^{2}$ | $3$ | $6$ | $14$ | $( 1,13, 8)( 2,14, 7)( 3,12,10,17,16, 5)( 4,11, 9,18,15, 6)$ |
6E-1 | $6^{2},3^{2}$ | $3$ | $6$ | $14$ | $( 1, 8,13)( 2, 7,14)( 3, 5,16,17,10,12)( 4, 6,15,18, 9,11)$ |
Malle's constant $a(G)$: $1/6$
magma: ConjugacyClasses(G);
Character table
1A | 2A | 2B | 2C | 3A1 | 3A-1 | 3B | 3C1 | 3C-1 | 6A1 | 6A-1 | 6B | 6C1 | 6C-1 | 6D1 | 6D-1 | 6E1 | 6E-1 | ||
Size | 1 | 1 | 3 | 3 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | |
2 P | 1A | 1A | 1A | 1A | 3A-1 | 3A1 | 3B | 3C-1 | 3C1 | 3A-1 | 3A1 | 3C-1 | 3B | 3C1 | 3A1 | 3A-1 | 3A1 | 3A-1 | |
3 P | 1A | 2A | 2B | 2C | 1A | 1A | 1A | 1A | 1A | 2A | 2A | 2A | 2A | 2A | 2B | 2B | 2C | 2C | |
Type | |||||||||||||||||||
36.12.1a | R | ||||||||||||||||||
36.12.1b | R | ||||||||||||||||||
36.12.1c | R | ||||||||||||||||||
36.12.1d | R | ||||||||||||||||||
36.12.1e1 | C | ||||||||||||||||||
36.12.1e2 | C | ||||||||||||||||||
36.12.1f1 | C | ||||||||||||||||||
36.12.1f2 | C | ||||||||||||||||||
36.12.1g1 | C | ||||||||||||||||||
36.12.1g2 | C | ||||||||||||||||||
36.12.1h1 | C | ||||||||||||||||||
36.12.1h2 | C | ||||||||||||||||||
36.12.2a | R | ||||||||||||||||||
36.12.2b | R | ||||||||||||||||||
36.12.2c1 | C | ||||||||||||||||||
36.12.2c2 | C | ||||||||||||||||||
36.12.2d1 | C | ||||||||||||||||||
36.12.2d2 | C |
magma: CharacterTable(G);
Regular extensions
Data not computed