Show commands:
Magma
magma: G := TransitiveGroup(18, 14);
Group action invariants
Degree $n$: | $18$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $14$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_9:C_6$ | ||
Parity: | $-1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $6$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,17,15,13,11,9,8,6,3,2,18,16,14,12,10,7,5,4), (1,3,11,8,10,18,14,15,5)(2,4,12,7,9,17,13,16,6) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $3$: $C_3$ x 4 $6$: $C_6$ x 4 $9$: $C_3^2$ $18$: $C_6 \times C_3$ $27$: $C_9:C_3$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: $C_3$
Degree 6: $C_6$
Degree 9: $C_9:C_3$
Low degree siblings
There are no siblings with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
Label | Cycle Type | Size | Order | Representative |
$1^{18}$ | $1$ | $1$ | $()$ | |
$3^{4},1^{6}$ | $3$ | $3$ | $( 3,10,15)( 4, 9,16)( 5,18,11)( 6,17,12)$ | |
$3^{4},1^{6}$ | $3$ | $3$ | $( 3,15,10)( 4,16, 9)( 5,11,18)( 6,12,17)$ | |
$2^{9}$ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)$ | |
$6^{2},2^{3}$ | $3$ | $6$ | $( 1, 2)( 3, 9,15, 4,10,16)( 5,17,11, 6,18,12)( 7, 8)(13,14)$ | |
$6^{2},2^{3}$ | $3$ | $6$ | $( 1, 2)( 3,16,10, 4,15, 9)( 5,12,18, 6,11,17)( 7, 8)(13,14)$ | |
$9^{2}$ | $3$ | $9$ | $( 1, 3, 5, 8,10,11,14,15,18)( 2, 4, 6, 7, 9,12,13,16,17)$ | |
$9^{2}$ | $3$ | $9$ | $( 1, 3,11, 8,10,18,14,15, 5)( 2, 4,12, 7, 9,17,13,16, 6)$ | |
$9^{2}$ | $3$ | $9$ | $( 1, 3,18, 8,10, 5,14,15,11)( 2, 4,17, 7, 9, 6,13,16,12)$ | |
$18$ | $3$ | $18$ | $( 1, 4, 5, 7,10,12,14,16,18, 2, 3, 6, 8, 9,11,13,15,17)$ | |
$18$ | $3$ | $18$ | $( 1, 4,11, 7,10,17,14,16, 5, 2, 3,12, 8, 9,18,13,15, 6)$ | |
$18$ | $3$ | $18$ | $( 1, 4,18, 7,10, 6,14,16,11, 2, 3,17, 8, 9, 5,13,15,12)$ | |
$9^{2}$ | $3$ | $9$ | $( 1, 5,15,14,18,10, 8,11, 3)( 2, 6,16,13,17, 9, 7,12, 4)$ | |
$9^{2}$ | $3$ | $9$ | $( 1, 5,10,14,18, 3, 8,11,15)( 2, 6, 9,13,17, 4, 7,12,16)$ | |
$9^{2}$ | $3$ | $9$ | $( 1, 5, 3,14,18,15, 8,11,10)( 2, 6, 4,13,17,16, 7,12, 9)$ | |
$18$ | $3$ | $18$ | $( 1, 6,15,13,18, 9, 8,12, 3, 2, 5,16,14,17,10, 7,11, 4)$ | |
$18$ | $3$ | $18$ | $( 1, 6,10,13,18, 4, 8,12,15, 2, 5, 9,14,17, 3, 7,11,16)$ | |
$18$ | $3$ | $18$ | $( 1, 6, 3,13,18,16, 8,12,10, 2, 5, 4,14,17,15, 7,11, 9)$ | |
$6^{3}$ | $1$ | $6$ | $( 1, 7,14, 2, 8,13)( 3, 9,15, 4,10,16)( 5,12,18, 6,11,17)$ | |
$3^{6}$ | $1$ | $3$ | $( 1, 8,14)( 2, 7,13)( 3,10,15)( 4, 9,16)( 5,11,18)( 6,12,17)$ | |
$6^{3}$ | $1$ | $6$ | $( 1,13, 8, 2,14, 7)( 3,16,10, 4,15, 9)( 5,17,11, 6,18,12)$ | |
$3^{6}$ | $1$ | $3$ | $( 1,14, 8)( 2,13, 7)( 3,15,10)( 4,16, 9)( 5,18,11)( 6,17,12)$ |
magma: ConjugacyClasses(G);
Group invariants
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: | $2$ | ||
Label: | 54.11 | magma: IdentifyGroup(G);
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Character table: |
1A | 2A | 3A1 | 3A-1 | 3B1 | 3B-1 | 6A1 | 6A-1 | 6B1 | 6B-1 | 9A1 | 9A-1 | 9B1 | 9B-1 | 9C1 | 9C-1 | 18A1 | 18A-1 | 18B1 | 18B-1 | 18C1 | 18C-1 | ||
Size | 1 | 1 | 1 | 1 | 3 | 3 | 1 | 1 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | |
2 P | 1A | 1A | 3A-1 | 3A1 | 3B-1 | 3B1 | 3A1 | 3A-1 | 3B1 | 3B-1 | 9C-1 | 9B1 | 9A-1 | 9B-1 | 9A1 | 9C1 | 9C1 | 9B1 | 9A-1 | 9B-1 | 9A1 | 9C-1 | |
3 P | 1A | 2A | 1A | 1A | 1A | 1A | 2A | 2A | 2A | 2A | 3A-1 | 3A-1 | 3A1 | 3A1 | 3A-1 | 3A1 | 6A-1 | 6A1 | 6A-1 | 6A-1 | 6A1 | 6A1 | |
Type | |||||||||||||||||||||||
54.11.1a | R | ||||||||||||||||||||||
54.11.1b | R | ||||||||||||||||||||||
54.11.1c1 | C | ||||||||||||||||||||||
54.11.1c2 | C | ||||||||||||||||||||||
54.11.1d1 | C | ||||||||||||||||||||||
54.11.1d2 | C | ||||||||||||||||||||||
54.11.1e1 | C | ||||||||||||||||||||||
54.11.1e2 | C | ||||||||||||||||||||||
54.11.1f1 | C | ||||||||||||||||||||||
54.11.1f2 | C | ||||||||||||||||||||||
54.11.1g1 | C | ||||||||||||||||||||||
54.11.1g2 | C | ||||||||||||||||||||||
54.11.1h1 | C | ||||||||||||||||||||||
54.11.1h2 | C | ||||||||||||||||||||||
54.11.1i1 | C | ||||||||||||||||||||||
54.11.1i2 | C | ||||||||||||||||||||||
54.11.1j1 | C | ||||||||||||||||||||||
54.11.1j2 | C | ||||||||||||||||||||||
54.11.3a1 | C | ||||||||||||||||||||||
54.11.3a2 | C | ||||||||||||||||||||||
54.11.3b1 | C | ||||||||||||||||||||||
54.11.3b2 | C |
magma: CharacterTable(G);