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Magma
magma: G := TransitiveGroup(18, 49);
Group invariants
Abstract group: | $\He_3:C_4$ | 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 $t$: | $49$ | 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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$\card{\Aut(F/K)}$: | $3$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | $(1,8,14)(2,9,15)(3,7,13)(4,10,18)(5,11,16)(6,12,17)$, $(1,6,15,10,3,5,14,12,2,4,13,11)(7,17,8,18,9,16)$ | magma: Generators(G);
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Low degree resolvents
$\card{(G/N)}$ Galois groups for stem field(s) $2$: $C_2$ $4$: $C_4$ $36$: $C_3^2:C_4$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: None
Degree 6: $C_3^2:C_4$
Degree 9: None
Low degree siblings
18T49, 27T32, 36T85 x 2Siblings 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 | $2^{6},1^{6}$ | $9$ | $2$ | $6$ | $( 1,14)( 2,15)( 3,13)( 4,10)( 5,11)( 6,12)$ |
3A1 | $3^{6}$ | $1$ | $3$ | $12$ | $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,12,11)(13,15,14)(16,18,17)$ |
3A-1 | $3^{6}$ | $1$ | $3$ | $12$ | $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)$ |
3B | $3^{5},1^{3}$ | $12$ | $3$ | $10$ | $( 1,13, 7)( 2,14, 8)( 3,15, 9)( 4, 5, 6)(16,18,17)$ |
3C | $3^{6}$ | $12$ | $3$ | $12$ | $( 1,14, 7)( 2,15, 8)( 3,13, 9)( 4,10,16)( 5,11,17)( 6,12,18)$ |
4A1 | $4^{3},2^{3}$ | $9$ | $4$ | $12$ | $( 1, 4,14,10)( 2, 5,15,11)( 3, 6,13,12)( 7,18)( 8,16)( 9,17)$ |
4A-1 | $4^{3},2^{3}$ | $9$ | $4$ | $12$ | $( 1,10,14, 4)( 2,11,15, 5)( 3,12,13, 6)( 7,18)( 8,16)( 9,17)$ |
6A1 | $6^{2},3^{2}$ | $9$ | $6$ | $14$ | $( 1,15, 3,14, 2,13)( 4,11, 6,10, 5,12)( 7, 8, 9)(16,17,18)$ |
6A-1 | $6^{2},3^{2}$ | $9$ | $6$ | $14$ | $( 1,13, 2,14, 3,15)( 4,12, 5,10, 6,11)( 7, 9, 8)(16,18,17)$ |
12A1 | $12,6$ | $9$ | $12$ | $16$ | $( 1, 6,15,10, 3, 5,14,12, 2, 4,13,11)( 7,17, 8,18, 9,16)$ |
12A-1 | $12,6$ | $9$ | $12$ | $16$ | $( 1,11,13, 4, 2,12,14, 5, 3,10,15, 6)( 7,16, 9,18, 8,17)$ |
12A5 | $12,6$ | $9$ | $12$ | $16$ | $( 1, 5,13,10, 2, 6,14,11, 3, 4,15,12)( 7,16, 9,18, 8,17)$ |
12A-5 | $12,6$ | $9$ | $12$ | $16$ | $( 1,12,15, 4, 3,11,14, 6, 2,10,13, 5)( 7,17, 8,18, 9,16)$ |
Malle's constant $a(G)$: $1/6$
magma: ConjugacyClasses(G);
Character table
1A | 2A | 3A1 | 3A-1 | 3B | 3C | 4A1 | 4A-1 | 6A1 | 6A-1 | 12A1 | 12A-1 | 12A5 | 12A-5 | ||
Size | 1 | 9 | 1 | 1 | 12 | 12 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | |
2 P | 1A | 1A | 3A-1 | 3A1 | 3B | 3C | 2A | 2A | 3A1 | 3A-1 | 6A1 | 6A-1 | 6A-1 | 6A1 | |
3 P | 1A | 2A | 1A | 1A | 1A | 1A | 4A-1 | 4A1 | 2A | 2A | 4A-1 | 4A1 | 4A-1 | 4A1 | |
Type | |||||||||||||||
108.15.1a | R | ||||||||||||||
108.15.1b | R | ||||||||||||||
108.15.1c1 | C | ||||||||||||||
108.15.1c2 | C | ||||||||||||||
108.15.3a1 | C | ||||||||||||||
108.15.3a2 | C | ||||||||||||||
108.15.3b1 | C | ||||||||||||||
108.15.3b2 | C | ||||||||||||||
108.15.3c1 | C | ||||||||||||||
108.15.3c2 | C | ||||||||||||||
108.15.3c3 | C | ||||||||||||||
108.15.3c4 | C | ||||||||||||||
108.15.4a | R | ||||||||||||||
108.15.4b | R |
magma: CharacterTable(G);
Regular extensions
Data not computed