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Properties

Label 18T12
Degree $18$
Order $36$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_6:S_3$

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Show commands: Magma

magma: G := TransitiveGroup(18, 12);
 

Group action invariants

Degree $n$:  $18$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $12$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_6:S_3$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $2$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,17)(2,18)(3,4)(5,13)(6,14)(7,12)(8,11)(9,16)(10,15), (1,15)(2,16)(3,13)(4,14)(5,6)(7,9)(8,10)(11,17)(12,18), (1,11)(2,12)(3,15)(4,16)(5,8)(6,7)(13,18)(14,17)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$6$:  $S_3$ x 4
$12$:  $D_{6}$ x 4
$18$:  $C_3^2:C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$ x 4

Degree 6: $D_{6}$ x 4

Degree 9: $C_3^2:C_2$

Low degree siblings

18T12, 36T8

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

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

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:  36.13
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A 3B 3C 3D 6A 6B 6C 6D
Size 1 1 9 9 2 2 2 2 2 2 2 2
2 P 1A 1A 1A 1A 3D 3A 3C 3B 3D 3C 3B 3A
3 P 1A 2A 2B 2C 1A 1A 1A 1A 2A 2A 2A 2A
Type
36.13.1a R 1 1 1 1 1 1 1 1 1 1 1 1
36.13.1b R 1 −1 −1 1 1 1 1 1 −1 −1 −1 −1
36.13.1c R 1 −1 1 −1 1 1 1 1 −1 −1 −1 −1
36.13.1d R 1 1 −1 −1 1 1 1 1 1 1 1 1
36.13.2a R 2 2 0 0 −1 −1 −1 2 −1 −1 −1 2
36.13.2b R 2 2 0 0 −1 −1 2 −1 −1 −1 2 −1
36.13.2c R 2 2 0 0 −1 2 −1 −1 −1 2 −1 −1
36.13.2d R 2 2 0 0 2 −1 −1 −1 2 −1 −1 −1
36.13.2e R 2 −2 0 0 −1 −1 −1 2 1 1 1 −2
36.13.2f R 2 −2 0 0 −1 −1 2 −1 1 1 −2 1
36.13.2g R 2 −2 0 0 −1 2 −1 −1 1 −2 1 1
36.13.2h R 2 −2 0 0 2 −1 −1 −1 −2 1 1 1

magma: CharacterTable(G);