Properties

Label 12T18
Degree $12$
Order $36$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_6\times S_3$

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Show commands: Magma

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

Group action invariants

Degree $n$:  $12$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $18$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_6\times S_3$
CHM label:   $[3^{2}]E(4)$
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:  (1,10)(2,5)(3,12)(4,7)(6,9)(8,11), (2,6,10)(4,8,12), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$3$:  $C_3$
$4$:  $C_2^2$
$6$:  $S_3$, $C_6$ x 3
$12$:  $D_{6}$, $C_6\times C_2$
$18$:  $S_3\times C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 3: None

Degree 4: $C_2^2$

Degree 6: $S_3\times C_3$

Low degree siblings

18T6 x 2, 36T6

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderRepresentative
$1^{12}$ $1$ $1$ $()$
$3^{2},1^{6}$ $2$ $3$ $( 2, 6,10)( 4, 8,12)$
$3^{2},1^{6}$ $2$ $3$ $( 2,10, 6)( 4,12, 8)$
$2^{6}$ $3$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)$
$6^{2}$ $3$ $6$ $( 1, 2, 5, 6, 9,10)( 3, 4, 7, 8,11,12)$
$6^{2}$ $3$ $6$ $( 1, 2, 9,10, 5, 6)( 3, 4,11,12, 7, 8)$
$6^{2}$ $1$ $6$ $( 1, 3, 5, 7, 9,11)( 2, 4, 6, 8,10,12)$
$6,2^{3}$ $2$ $6$ $( 1, 3, 5, 7, 9,11)( 2, 8)( 4,10)( 6,12)$
$6^{2}$ $2$ $6$ $( 1, 3, 5, 7, 9,11)( 2,12,10, 8, 6, 4)$
$6^{2}$ $3$ $6$ $( 1, 4, 5, 8, 9,12)( 2, 3, 6, 7,10,11)$
$6^{2}$ $3$ $6$ $( 1, 4, 9,12, 5, 8)( 2, 7,10, 3, 6,11)$
$2^{6}$ $3$ $2$ $( 1, 4)( 2,11)( 3, 6)( 5, 8)( 7,10)( 9,12)$
$3^{4}$ $1$ $3$ $( 1, 5, 9)( 2, 6,10)( 3, 7,11)( 4, 8,12)$
$3^{4}$ $2$ $3$ $( 1, 5, 9)( 2,10, 6)( 3, 7,11)( 4,12, 8)$
$2^{6}$ $1$ $2$ $( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12)$
$6,2^{3}$ $2$ $6$ $( 1, 7)( 2,12,10, 8, 6, 4)( 3, 9)( 5,11)$
$3^{4}$ $1$ $3$ $( 1, 9, 5)( 2,10, 6)( 3,11, 7)( 4,12, 8)$
$6^{2}$ $1$ $6$ $( 1,11, 9, 7, 5, 3)( 2,12,10, 8, 6, 4)$

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:  36.12
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A1 3A-1 3B 3C1 3C-1 6A1 6A-1 6B 6C1 6C-1 6D1 6D-1 6E1 6E-1
Size 1 1 3 3 1 1 2 2 2 1 1 2 2 2 3 3 3 3
2 P 1A 1A 1A 1A 3A-1 3A1 3B 3C-1 3C1 3A-1 3A1 3C-1 3B 3C1 3A1 3A-1 3A1 3A-1
3 P 1A 2A 2B 2C 1A 1A 1A 1A 1A 2A 2A 2A 2A 2A 2B 2B 2C 2C
Type
36.12.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
36.12.1b R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
36.12.1c R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
36.12.1d R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
36.12.1e1 C 1 1 1 1 ζ31 ζ3 1 ζ31 ζ3 ζ31 ζ3 1 ζ3 ζ31 ζ3 ζ31 ζ3 ζ31
36.12.1e2 C 1 1 1 1 ζ3 ζ31 1 ζ3 ζ31 ζ3 ζ31 1 ζ31 ζ3 ζ31 ζ3 ζ31 ζ3
36.12.1f1 C 1 1 1 1 ζ31 ζ3 1 ζ31 ζ3 ζ31 ζ3 1 ζ3 ζ31 ζ3 ζ31 ζ3 ζ31
36.12.1f2 C 1 1 1 1 ζ3 ζ31 1 ζ3 ζ31 ζ3 ζ31 1 ζ31 ζ3 ζ31 ζ3 ζ31 ζ3
36.12.1g1 C 1 1 1 1 ζ31 ζ3 1 ζ31 ζ3 ζ31 ζ3 1 ζ3 ζ31 ζ3 ζ31 ζ3 ζ31
36.12.1g2 C 1 1 1 1 ζ3 ζ31 1 ζ3 ζ31 ζ3 ζ31 1 ζ31 ζ3 ζ31 ζ3 ζ31 ζ3
36.12.1h1 C 1 1 1 1 ζ31 ζ3 1 ζ31 ζ3 ζ31 ζ3 1 ζ3 ζ31 ζ3 ζ31 ζ3 ζ31
36.12.1h2 C 1 1 1 1 ζ3 ζ31 1 ζ3 ζ31 ζ3 ζ31 1 ζ31 ζ3 ζ31 ζ3 ζ31 ζ3
36.12.2a R 2 2 0 0 2 2 1 1 1 2 2 1 1 1 0 0 0 0
36.12.2b R 2 2 0 0 2 2 1 1 1 2 2 1 1 1 0 0 0 0
36.12.2c1 C 2 2 0 0 2ζ31 2ζ3 1 ζ31 ζ3 2ζ31 2ζ3 1 ζ3 ζ31 0 0 0 0
36.12.2c2 C 2 2 0 0 2ζ3 2ζ31 1 ζ3 ζ31 2ζ3 2ζ31 1 ζ31 ζ3 0 0 0 0
36.12.2d1 C 2 2 0 0 2ζ31 2ζ3 1 ζ31 ζ3 2ζ31 2ζ3 1 ζ3 ζ31 0 0 0 0
36.12.2d2 C 2 2 0 0 2ζ3 2ζ31 1 ζ3 ζ31 2ζ3 2ζ31 1 ζ31 ζ3 0 0 0 0

magma: CharacterTable(G);