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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 invariants

Abstract group:  $C_3^2.A_4$
magma: IdentifyGroup(G);
 
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
magma: NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $18$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $47$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(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

$\card{(G/N)}$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 TypeSizeOrderIndexRepresentative
1A $1^{18}$ $1$ $1$ $0$ $()$
2A $2^{6},1^{6}$ $3$ $2$ $6$ $( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)$
3A1 $3^{6}$ $1$ $3$ $12$ $( 1, 5, 4)( 2, 6, 3)( 7,12,10)( 8,11, 9)(13,18,15)(14,17,16)$
3A-1 $3^{6}$ $1$ $3$ $12$ $( 1, 4, 5)( 2, 3, 6)( 7,10,12)( 8, 9,11)(13,15,18)(14,16,17)$
3B1 $3^{4},1^{6}$ $3$ $3$ $8$ $( 7,12,10)( 8,11, 9)(13,15,18)(14,16,17)$
3B-1 $3^{4},1^{6}$ $3$ $3$ $8$ $( 7,10,12)( 8, 9,11)(13,18,15)(14,17,16)$
6A1 $6^{2},1^{6}$ $3$ $6$ $10$ $( 7, 9,12, 8,10,11)(13,17,15,14,18,16)$
6A-1 $6^{2},1^{6}$ $3$ $6$ $10$ $( 1, 3, 5, 2, 4, 6)(13,17,15,14,18,16)$
6B1 $6^{2},3^{2}$ $3$ $6$ $14$ $( 1, 3, 5, 2, 4, 6)( 7,10,12)( 8, 9,11)(13,16,18,14,15,17)$
6B-1 $6^{2},3^{2}$ $3$ $6$ $14$ $( 1, 6, 4, 2, 5, 3)( 7,12,10)( 8,11, 9)(13,17,15,14,18,16)$
6C1 $6,3^{2},2^{3}$ $3$ $6$ $12$ $( 1, 5, 4)( 2, 6, 3)( 7, 8)( 9,10)(11,12)(13,16,18,14,15,17)$
6C-1 $6,3^{2},2^{3}$ $3$ $6$ $12$ $( 1, 6, 4, 2, 5, 3)( 7,10,12)( 8, 9,11)(13,14)(15,16)(17,18)$
6D1 $6,3^{2},2^{3}$ $3$ $6$ $12$ $( 1, 4, 5)( 2, 3, 6)( 7,11,10, 8,12, 9)(13,14)(15,16)(17,18)$
6D-1 $6,3^{2},2^{3}$ $3$ $6$ $12$ $( 1, 2)( 3, 4)( 5, 6)( 7,12,10)( 8,11, 9)(13,16,18,14,15,17)$
9A1 $9^{2}$ $12$ $9$ $16$ $( 1, 9,17, 5, 8,16, 4,11,14)( 2,10,18, 6, 7,15, 3,12,13)$
9A-1 $9^{2}$ $12$ $9$ $16$ $( 1,14,11, 4,16, 8, 5,17, 9)( 2,13,12, 3,15, 7, 6,18,10)$
9B1 $9^{2}$ $12$ $9$ $16$ $( 1,17,11, 4,14, 8, 5,16, 9)( 2,18,12, 3,13, 7, 6,15,10)$
9B-1 $9^{2}$ $12$ $9$ $16$ $( 1, 9,16, 5, 8,14, 4,11,17)( 2,10,15, 6, 7,13, 3,12,18)$
9C1 $9^{2}$ $12$ $9$ $16$ $( 1, 8,16, 5,11,14, 4, 9,17)( 2, 7,15, 6,12,13, 3,10,18)$
9C-1 $9^{2}$ $12$ $9$ $16$ $( 1,17, 9, 4,14,11, 5,16, 8)( 2,18,10, 3,13,12, 6,15, 7)$

Malle's constant $a(G)$:     $1/6$

magma: ConjugacyClasses(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 3A1 3A-1 3B1 3B-1 3B1 3B-1 9A-1 9A1 9B-1 9B1 9C-1 9C1
3 P 1A 2A 1A 1A 1A 1A 2A 2A 2A 2A 2A 2A 2A 2A 3A1 3A-1 3A-1 3A1 3A1 3A-1
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 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 1 1 ζ3 ζ3−1 ζ3 ζ3 1 1 ζ3−1
108.21.1b2 C 1 1 1 1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 1 1 ζ3−1 ζ3 ζ3−1 ζ3−1 1 1 ζ3
108.21.1c1 C 1 1 1 1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 1 1 ζ3 ζ3 ζ3−1 1 ζ3−1 ζ3 1
108.21.1c2 C 1 1 1 1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 1 1 ζ3−1 ζ3−1 ζ3 1 ζ3 ζ3−1 1
108.21.1d1 C 1 1 1 1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 1 1 ζ3 1 1 ζ3−1 ζ3 ζ3−1 ζ3
108.21.1d2 C 1 1 1 1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 1 1 ζ3−1 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 ζ3−1 ζ3−1 ζ3 ζ3
108.21.1e2 C 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ζ3 ζ3−1 ζ3 ζ3 ζ3−1 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 0 0 0 0 0 3ζ3 3ζ3−1 0 0 0 0 0 0 0
108.21.3b2 C 3 3 3ζ3 3ζ3−1 0 0 0 0 0 0 0 3ζ3−1 3ζ3 0 0 0 0 0 0 0
108.21.3c1 C 3 −1 3 3 3ζ3−1 3ζ3 −ζ3−1 −ζ3 −ζ3−1 −ζ3 −ζ3−1 −1 −1 −ζ3 0 0 0 0 0 0
108.21.3c2 C 3 −1 3 3 3ζ3 3ζ3−1 −ζ3 −ζ3−1 −ζ3 −ζ3−1 −ζ3 −1 −1 −ζ3−1 0 0 0 0 0 0
108.21.3d1 C 3 −1 3ζ3−1 3ζ3 0 0 2ζ3−1 2ζ3−1 2ζ3 2ζ3 2 −ζ3 −ζ3−1 2 0 0 0 0 0 0
108.21.3d2 C 3 −1 3ζ3 3ζ3−1 0 0 2ζ3 2ζ3 2ζ3−1 2ζ3−1 2 −ζ3−1 −ζ3 2 0 0 0 0 0 0
108.21.3e1 C 3 −1 3ζ3−1 3ζ3 0 0 2ζ3 2 2 2ζ3−1 2ζ3−1 −ζ3 −ζ3−1 2ζ3 0 0 0 0 0 0
108.21.3e2 C 3 −1 3ζ3 3ζ3−1 0 0 2ζ3−1 2 2 2ζ3 2ζ3 −ζ3−1 −ζ3 2ζ3−1 0 0 0 0 0 0
108.21.3f1 C 3 −1 3ζ3−1 3ζ3 0 0 2 2ζ3 2ζ3−1 2 2ζ3 −ζ3 −ζ3−1 2ζ3−1 0 0 0 0 0 0
108.21.3f2 C 3 −1 3ζ3 3ζ3−1 0 0 2 2ζ3−1 2ζ3 2 2ζ3−1 −ζ3−1 −ζ3 2ζ3 0 0 0 0 0 0

magma: CharacterTable(G);
 

Regular extensions

Data not computed