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Properties

Label 18T47
Degree $18$
Order $108$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3^2.A_4$

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Show commands: Magma

magma: G := TransitiveGroup(18, 47);
 

Group action invariants

Degree $n$:  $18$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $47$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_3^2.A_4$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $6$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,13,9,4,15,11,5,18,8)(2,14,10,3,16,12,6,17,7), (1,17,9,4,14,11,5,16,8)(2,18,10,3,13,12,6,15,7)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$3$:  $C_3$ x 4
$9$:  $C_3^2$
$12$:  $A_4$
$27$:  $C_9:C_3$
$36$:  $C_3\times A_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $C_3$

Degree 6: $A_4$

Degree 9: $C_9:C_3$

Low degree siblings

18T47 x 2, 36T83

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

LabelCycle TypeSizeOrderRepresentative
$1^{18}$ $1$ $1$ $()$
$2^{6},1^{6}$ $3$ $2$ $( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)$
$6^{2},1^{6}$ $3$ $6$ $( 7, 9,12, 8,10,11)(13,17,15,14,18,16)$
$3^{4},1^{6}$ $3$ $3$ $( 7,10,12)( 8, 9,11)(13,18,15)(14,17,16)$
$6^{2},1^{6}$ $3$ $6$ $( 7,11,10, 8,12, 9)(13,16,18,14,15,17)$
$3^{4},1^{6}$ $3$ $3$ $( 7,12,10)( 8,11, 9)(13,15,18)(14,16,17)$
$6,3^{2},2^{3}$ $3$ $6$ $( 1, 2)( 3, 4)( 5, 6)( 7, 9,12, 8,10,11)(13,18,15)(14,17,16)$
$6,3^{2},2^{3}$ $3$ $6$ $( 1, 2)( 3, 4)( 5, 6)( 7,10,12)( 8, 9,11)(13,17,15,14,18,16)$
$6,3^{2},2^{3}$ $3$ $6$ $( 1, 2)( 3, 4)( 5, 6)( 7,11,10, 8,12, 9)(13,15,18)(14,16,17)$
$6,3^{2},2^{3}$ $3$ $6$ $( 1, 2)( 3, 4)( 5, 6)( 7,12,10)( 8,11, 9)(13,16,18,14,15,17)$
$6^{2},3^{2}$ $3$ $6$ $( 1, 3, 5, 2, 4, 6)( 7, 9,12, 8,10,11)(13,15,18)(14,16,17)$
$3^{6}$ $1$ $3$ $( 1, 4, 5)( 2, 3, 6)( 7,10,12)( 8, 9,11)(13,15,18)(14,16,17)$
$6^{2},3^{2}$ $3$ $6$ $( 1, 5, 4)( 2, 6, 3)( 7,11,10, 8,12, 9)(13,17,15,14,18,16)$
$3^{6}$ $1$ $3$ $( 1, 5, 4)( 2, 6, 3)( 7,12,10)( 8,11, 9)(13,18,15)(14,17,16)$
$9^{2}$ $12$ $9$ $( 1, 7,13, 5,12,18, 4,10,15)( 2, 8,14, 6,11,17, 3, 9,16)$
$9^{2}$ $12$ $9$ $( 1, 7,15, 5,12,13, 4,10,18)( 2, 8,16, 6,11,14, 3, 9,17)$
$9^{2}$ $12$ $9$ $( 1, 7,17, 5,12,16, 4,10,14)( 2, 8,18, 6,11,15, 3, 9,13)$
$9^{2}$ $12$ $9$ $( 1,13,10, 4,15,12, 5,18, 7)( 2,14, 9, 3,16,11, 6,17, 8)$
$9^{2}$ $12$ $9$ $( 1,13, 8, 4,15, 9, 5,18,11)( 2,14, 7, 3,16,10, 6,17,12)$
$9^{2}$ $12$ $9$ $( 1,13,12, 4,15, 7, 5,18,10)( 2,14,11, 3,16, 8, 6,17, 9)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $108=2^{2} \cdot 3^{3}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  108.21
magma: IdentifyGroup(G);
 
Character table:

1A 2A 3A1 3A-1 3B1 3B-1 6A1 6A-1 6B1 6B-1 6C1 6C-1 6D1 6D-1 9A1 9A-1 9B1 9B-1 9C1 9C-1
Size 1 3 1 1 3 3 3 3 3 3 3 3 3 3 12 12 12 12 12 12
2 P 1A 1A 3A-1 3A1 3B-1 3B1 3B1 3B-1 3B1 3B-1 3A-1 3B-1 3B1 3A1 9C1 9C-1 9B-1 9A1 9A-1 9B1
3 P 1A 2A 1A 1A 1A 1A 2A 2A 2A 2A 2A 2A 2A 2A 3A-1 3A1 3A-1 3A-1 3A1 3A1
Type
108.21.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
108.21.1b1 C 1 1 1 1 ζ3−1 ζ3 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 1 1
108.21.1b2 C 1 1 1 1 ζ3 ζ3−1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 1 1
108.21.1c1 C 1 1 1 1 ζ3−1 ζ3 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 1 1 ζ3 ζ3−1
108.21.1c2 C 1 1 1 1 ζ3 ζ3−1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 1 1 ζ3−1 ζ3
108.21.1d1 C 1 1 1 1 ζ3−1 ζ3 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 1 1 ζ3−1 ζ3 ζ3−1 ζ3
108.21.1d2 C 1 1 1 1 ζ3 ζ3−1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 1 1 ζ3 ζ3−1 ζ3 ζ3−1
108.21.1e1 C 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3
108.21.1e2 C 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1
108.21.3a R 3 −1 3 3 3 3 −1 −1 −1 −1 −1 −1 −1 −1 0 0 0 0 0 0
108.21.3b1 C 3 3 3ζ3−1 3ζ3 0 0 3ζ3 3ζ3−1 0 0 0 0 0 0 0 0 0 0 0 0
108.21.3b2 C 3 3 3ζ3 3ζ3−1 0 0 3ζ3−1 3ζ3 0 0 0 0 0 0 0 0 0 0 0 0
108.21.3c1 C 3 −1 3 3 3ζ3−1 3ζ3 −1 −1 −ζ3 −ζ3−1 −ζ3−1 −ζ3 −ζ3 −ζ3−1 0 0 0 0 0 0
108.21.3c2 C 3 −1 3 3 3ζ3 3ζ3−1 −1 −1 −ζ3−1 −ζ3 −ζ3 −ζ3−1 −ζ3−1 −ζ3 0 0 0 0 0 0
108.21.3d1 C 3 −1 3ζ3−1 3ζ3 0 0 −ζ3 −ζ3−1 2ζ3−1 2ζ3 2ζ3−1 2ζ3 2 2 0 0 0 0 0 0
108.21.3d2 C 3 −1 3ζ3 3ζ3−1 0 0 −ζ3−1 −ζ3 2ζ3 2ζ3−1 2ζ3 2ζ3−1 2 2 0 0 0 0 0 0
108.21.3e1 C 3 −1 3ζ3−1 3ζ3 0 0 −ζ3 −ζ3−1 2ζ3 2ζ3−1 2 2 2ζ3−1 2ζ3 0 0 0 0 0 0
108.21.3e2 C 3 −1 3ζ3 3ζ3−1 0 0 −ζ3−1 −ζ3 2ζ3−1 2ζ3 2 2 2ζ3 2ζ3−1 0 0 0 0 0 0
108.21.3f1 C 3 −1 3ζ3−1 3ζ3 0 0 −ζ3 −ζ3−1 2 2 2ζ3 2ζ3−1 2ζ3 2ζ3−1 0 0 0 0 0 0
108.21.3f2 C 3 −1 3ζ3 3ζ3−1 0 0 −ζ3−1 −ζ3 2 2 2ζ3−1 2ζ3 2ζ3−1 2ζ3 0 0 0 0 0 0

magma: CharacterTable(G);