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Magma
magma: G := TransitiveGroup(18, 45);
Group invariants
Abstract group: | $C_{18}:C_6$ | magma: IdentifyGroup(G);
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Order: | $108=2^{2} \cdot 3^{3}$ | 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 $t$: | $45$ | 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: | $(3,6,10,17,15,12)(4,5,9,18,16,11)(7,13)(8,14)$, $(1,18,14,5,8,11)(2,17,13,6,7,12)(3,4)(9,15)(10,16)$ | 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 $12$: $D_{6}$, $C_6\times C_2$ $18$: $S_3\times C_3$ $36$: $C_6\times S_3$ $54$: $(C_9:C_3):C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: $S_3$
Degree 6: $D_{6}$
Degree 9: $(C_9:C_3):C_2$
Low degree siblings
18T45, 36T67Siblings 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 | $2^{9}$ | $9$ | $2$ | $9$ | $( 1,11)( 2,12)( 3, 9)( 4,10)( 5, 8)( 6, 7)(13,17)(14,18)(15,16)$ |
2C | $2^{8},1^{2}$ | $9$ | $2$ | $8$ | $( 3,17)( 4,18)( 5,16)( 6,15)( 7,13)( 8,14)( 9,11)(10,12)$ |
3A | $3^{6}$ | $2$ | $3$ | $12$ | $( 1,14, 8)( 2,13, 7)( 3,15,10)( 4,16, 9)( 5,18,11)( 6,17,12)$ |
3B1 | $3^{4},1^{6}$ | $3$ | $3$ | $8$ | $( 3,15,10)( 4,16, 9)( 5,11,18)( 6,12,17)$ |
3B-1 | $3^{4},1^{6}$ | $3$ | $3$ | $8$ | $( 3,10,15)( 4, 9,16)( 5,18,11)( 6,17,12)$ |
6A | $6^{3}$ | $2$ | $6$ | $15$ | $( 1, 7,14, 2, 8,13)( 3, 9,15, 4,10,16)( 5,12,18, 6,11,17)$ |
6B1 | $6^{2},2^{3}$ | $3$ | $6$ | $13$ | $( 1, 7,14, 2, 8,13)( 3,16,10, 4,15, 9)( 5, 6)(11,12)(17,18)$ |
6B-1 | $6^{2},2^{3}$ | $3$ | $6$ | $13$ | $( 1, 7,14, 2, 8,13)( 3, 4)( 5,17,11, 6,18,12)( 9,10)(15,16)$ |
6C1 | $6^{2},2^{2},1^{2}$ | $9$ | $6$ | $12$ | $( 3, 6,10,17,15,12)( 4, 5, 9,18,16,11)( 7,13)( 8,14)$ |
6C-1 | $6^{2},2^{3}$ | $9$ | $6$ | $13$ | $( 1,18,14, 5, 8,11)( 2,17,13, 6, 7,12)( 3, 4)( 9,15)(10,16)$ |
6D1 | $6^{2},2^{3}$ | $9$ | $6$ | $13$ | $( 1, 5, 8,18,14,11)( 2, 6, 7,17,13,12)( 3,16)( 4,15)( 9,10)$ |
6D-1 | $6^{2},2^{2},1^{2}$ | $9$ | $6$ | $12$ | $( 3,12,15,17,10, 6)( 4,11,16,18, 9, 5)( 7,13)( 8,14)$ |
9A | $9^{2}$ | $6$ | $9$ | $16$ | $( 1,17,15,14,12,10, 8, 6, 3)( 2,18,16,13,11, 9, 7, 5, 4)$ |
9B1 | $9^{2}$ | $6$ | $9$ | $16$ | $( 1, 6,15,14,17,10, 8,12, 3)( 2, 5,16,13,18, 9, 7,11, 4)$ |
9B-1 | $9^{2}$ | $6$ | $9$ | $16$ | $( 1,12,15,14, 6,10, 8,17, 3)( 2,11,16,13, 5, 9, 7,18, 4)$ |
18A | $18$ | $6$ | $18$ | $17$ | $( 1, 9,17, 7,15, 5,14, 4,12, 2,10,18, 8,16, 6,13, 3,11)$ |
18B1 | $18$ | $6$ | $18$ | $17$ | $( 1,16,17, 7, 3, 5,14, 9,12, 2,15,18, 8, 4, 6,13,10,11)$ |
18B-1 | $18$ | $6$ | $18$ | $17$ | $( 1, 4,17, 7,10, 5,14,16,12, 2, 3,18, 8, 9, 6,13,15,11)$ |
Malle's constant $a(G)$: $1/8$
magma: ConjugacyClasses(G);
Character table
1A | 2A | 2B | 2C | 3A | 3B1 | 3B-1 | 6A | 6B1 | 6B-1 | 6C1 | 6C-1 | 6D1 | 6D-1 | 9A | 9B1 | 9B-1 | 18A | 18B1 | 18B-1 | ||
Size | 1 | 1 | 9 | 9 | 2 | 3 | 3 | 2 | 3 | 3 | 9 | 9 | 9 | 9 | 6 | 6 | 6 | 6 | 6 | 6 | |
2 P | 1A | 1A | 1A | 1A | 3A | 3B-1 | 3B1 | 3A | 3B1 | 3B-1 | 3B-1 | 3B-1 | 3B1 | 3B1 | 9A | 9B-1 | 9B1 | 9A | 9B1 | 9B-1 | |
3 P | 1A | 2A | 2B | 2C | 1A | 1A | 1A | 2A | 2A | 2A | 2C | 2B | 2B | 2C | 3A | 3A | 3A | 6A | 6A | 6A | |
Type | |||||||||||||||||||||
108.26.1a | R | ||||||||||||||||||||
108.26.1b | R | ||||||||||||||||||||
108.26.1c | R | ||||||||||||||||||||
108.26.1d | R | ||||||||||||||||||||
108.26.1e1 | C | ||||||||||||||||||||
108.26.1e2 | C | ||||||||||||||||||||
108.26.1f1 | C | ||||||||||||||||||||
108.26.1f2 | C | ||||||||||||||||||||
108.26.1g1 | C | ||||||||||||||||||||
108.26.1g2 | C | ||||||||||||||||||||
108.26.1h1 | C | ||||||||||||||||||||
108.26.1h2 | C | ||||||||||||||||||||
108.26.2a | R | ||||||||||||||||||||
108.26.2b | R | ||||||||||||||||||||
108.26.2c1 | C | ||||||||||||||||||||
108.26.2c2 | C | ||||||||||||||||||||
108.26.2d1 | C | ||||||||||||||||||||
108.26.2d2 | C | ||||||||||||||||||||
108.26.6a | R | ||||||||||||||||||||
108.26.6b | R |
magma: CharacterTable(G);
Regular extensions
Data not computed