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Magma
magma: G := TransitiveGroup(18, 23);
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 $t$: | $23$ | 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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$\card{\Aut(F/K)}$: | $9$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | $(1,18,7,6,14,10)(2,17,8,5,15,12)(3,16,9,4,13,11)$, $(1,8,13)(2,9,14)(3,7,15)(4,5,6)(10,11,12)(16,17,18)$, $(1,9,15)(2,7,13)(3,8,14)(4,6,5)(10,12,11)(16,18,17)$ | magma: Generators(G);
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Low degree resolvents
$\card{(G/N)}$ Galois groups for stem field(s) $2$: $C_2$ $3$: $C_3$ $6$: $S_3$ x 4, $C_6$ $18$: $S_3\times C_3$ x 4, $C_3^2:C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: $S_3$
Degree 6: $S_3$, $S_3\times C_3$ x 3
Degree 9: None
Low degree siblings
18T23 x 3, 27T13Siblings 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 | $2^{9}$ | $9$ | $2$ | $9$ | $( 1,10)( 2,12)( 3,11)( 4,13)( 5,15)( 6,14)( 7,18)( 8,17)( 9,16)$ |
3A1 | $3^{6}$ | $1$ | $3$ | $12$ | $( 1,14, 7)( 2,15, 8)( 3,13, 9)( 4,16,11)( 5,17,12)( 6,18,10)$ |
3A-1 | $3^{6}$ | $1$ | $3$ | $12$ | $( 1, 7,14)( 2, 8,15)( 3, 9,13)( 4,11,16)( 5,12,17)( 6,10,18)$ |
3B | $3^{6}$ | $2$ | $3$ | $12$ | $( 1,13, 8)( 2,14, 9)( 3,15, 7)( 4, 6, 5)(10,12,11)(16,18,17)$ |
3C | $3^{6}$ | $2$ | $3$ | $12$ | $( 1,13, 8)( 2,14, 9)( 3,15, 7)( 4,10,17)( 5,11,18)( 6,12,16)$ |
3D | $3^{6}$ | $2$ | $3$ | $12$ | $( 1, 9,15)( 2, 7,13)( 3, 8,14)( 4,18,12)( 5,16,10)( 6,17,11)$ |
3E | $3^{3},1^{9}$ | $2$ | $3$ | $6$ | $( 1, 7,14)( 2, 8,15)( 3, 9,13)$ |
3F1 | $3^{6}$ | $2$ | $3$ | $12$ | $( 1,15, 9)( 2,13, 7)( 3,14, 8)( 4,17,10)( 5,18,11)( 6,16,12)$ |
3F-1 | $3^{6}$ | $2$ | $3$ | $12$ | $( 1, 3, 2)( 4,18,12)( 5,16,10)( 6,17,11)( 7, 9, 8)(13,15,14)$ |
3G1 | $3^{6}$ | $2$ | $3$ | $12$ | $( 1, 8,13)( 2, 9,14)( 3, 7,15)( 4,12,18)( 5,10,16)( 6,11,17)$ |
3G-1 | $3^{6}$ | $2$ | $3$ | $12$ | $( 1, 9,15)( 2, 7,13)( 3, 8,14)( 4, 6, 5)(10,12,11)(16,18,17)$ |
3H1 | $3^{6}$ | $2$ | $3$ | $12$ | $( 1, 3, 2)( 4,10,17)( 5,11,18)( 6,12,16)( 7, 9, 8)(13,15,14)$ |
3H-1 | $3^{6}$ | $2$ | $3$ | $12$ | $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)$ |
3I1 | $3^{6}$ | $2$ | $3$ | $12$ | $( 1,14, 7)( 2,15, 8)( 3,13, 9)( 4,11,16)( 5,12,17)( 6,10,18)$ |
3I-1 | $3^{3},1^{9}$ | $2$ | $3$ | $6$ | $( 4,16,11)( 5,17,12)( 6,18,10)$ |
6A1 | $6^{3}$ | $9$ | $6$ | $15$ | $( 1,18,14,10, 7, 6)( 2,17,15,12, 8, 5)( 3,16,13,11, 9, 4)$ |
6A-1 | $6^{3}$ | $9$ | $6$ | $15$ | $( 1, 6, 7,10,14,18)( 2, 5, 8,12,15,17)( 3, 4, 9,11,13,16)$ |
Malle's constant $a(G)$: $1/6$
magma: ConjugacyClasses(G);
Character table
1A | 2A | 3A1 | 3A-1 | 3B | 3C | 3D | 3E | 3F1 | 3F-1 | 3G1 | 3G-1 | 3H1 | 3H-1 | 3I1 | 3I-1 | 6A1 | 6A-1 | ||
Size | 1 | 9 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 9 | 9 | |
2 P | 1A | 1A | 3A-1 | 3A1 | 3I-1 | 3C | 3B | 3H1 | 3F1 | 3G-1 | 3F-1 | 3G1 | 3I1 | 3E | 3D | 3H-1 | 3A1 | 3A-1 | |
3 P | 1A | 2A | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 2A | 2A | |
Type | |||||||||||||||||||
54.13.1a | R | ||||||||||||||||||
54.13.1b | R | ||||||||||||||||||
54.13.1c1 | C | ||||||||||||||||||
54.13.1c2 | C | ||||||||||||||||||
54.13.1d1 | C | ||||||||||||||||||
54.13.1d2 | C | ||||||||||||||||||
54.13.2a | R | ||||||||||||||||||
54.13.2b | R | ||||||||||||||||||
54.13.2c | R | ||||||||||||||||||
54.13.2d | R | ||||||||||||||||||
54.13.2e1 | C | ||||||||||||||||||
54.13.2e2 | C | ||||||||||||||||||
54.13.2f1 | C | ||||||||||||||||||
54.13.2f2 | C | ||||||||||||||||||
54.13.2g1 | C | ||||||||||||||||||
54.13.2g2 | C | ||||||||||||||||||
54.13.2h1 | C | ||||||||||||||||||
54.13.2h2 | C |
magma: CharacterTable(G);