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Properties

Label 18T8
Degree $18$
Order $36$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $A_4 \times C_3$

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Show commands: Magma

magma: G := TransitiveGroup(18, 8);
 

Group action invariants

Degree $n$:  $18$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $8$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $A_4 \times C_3$
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:  (1,16,10)(2,15,9)(3,18,11)(4,17,12)(5,13,8)(6,14,7), (1,8,17)(2,7,18)(3,10,14)(4,9,13)(5,11,15)(6,12,16)
magma: Generators(G);
 

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $C_3$ x 4

Degree 6: $A_4$

Degree 9: $C_3^2$

Low degree siblings

12T20 x 3, 36T12

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $36=2^{2} \cdot 3^{2}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  36.11
magma: IdentifyGroup(G);
 
Character table:

1A 2A 3A1 3A-1 3B1 3B-1 3C1 3C-1 3D1 3D-1 6A1 6A-1
Size 1 3 1 1 4 4 4 4 4 4 3 3
2 P 1A 1A 3A-1 3A1 3C-1 3B-1 3D-1 3C1 3D1 3B1 3A1 3A-1
3 P 1A 2A 1A 1A 1A 1A 1A 1A 1A 1A 2A 2A
Type
36.11.1a R 1 1 1 1 1 1 1 1 1 1 1 1
36.11.1b1 C 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1 1 1 ζ3 ζ3−1
36.11.1b2 C 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3 1 1 ζ3−1 ζ3
36.11.1c1 C 1 1 ζ3−1 ζ3 ζ3−1 ζ3 1 1 ζ3−1 ζ3 ζ3 ζ3−1
36.11.1c2 C 1 1 ζ3 ζ3−1 ζ3 ζ3−1 1 1 ζ3 ζ3−1 ζ3−1 ζ3
36.11.1d1 C 1 1 ζ3−1 ζ3 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1
36.11.1d2 C 1 1 ζ3 ζ3−1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3
36.11.1e1 C 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1 1 1
36.11.1e2 C 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3 1 1
36.11.3a R 3 −1 3 3 0 0 0 0 0 0 −1 −1
36.11.3b1 C 3 −1 3ζ3−1 3ζ3 0 0 0 0 0 0 −ζ3 −ζ3−1
36.11.3b2 C 3 −1 3ζ3 3ζ3−1 0 0 0 0 0 0 −ζ3−1 −ζ3

magma: CharacterTable(G);