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Properties

Label 12T43
Degree $12$
Order $72$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $S_3\times A_4$

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Show commands: Magma

magma: G := TransitiveGroup(12, 43);
 

Group action invariants

Degree $n$:  $12$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $43$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $S_3\times A_4$
CHM label:   $A(4)[x]S(3)$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $1$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,7,10)(2,5,11)(3,6,9), (2,8,11)(3,6,12)(4,7,10), (1,5)(2,10)(4,8)(7,11), (1,5,9)(2,6,10)(3,7,11)(4,8,12)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $S_3$, $C_6$
$12$:  $A_4$
$18$:  $S_3\times C_3$
$24$:  $A_4\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $S_3$

Degree 4: $A_4$

Degree 6: None

Low degree siblings

18T31, 18T32, 24T78, 24T83, 36T21, 36T50, 36T51

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 $ $1$ $1$ $()$
$ 6, 3, 2, 1 $ $12$ $6$ $( 2, 3, 8, 6,11,12)( 4,10, 7)( 5, 9)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $3$ $2$ $( 2, 6)( 3,11)( 5, 9)( 8,12)$
$ 3, 3, 3, 1, 1, 1 $ $4$ $3$ $( 2, 8,11)( 3, 6,12)( 4, 7,10)$
$ 3, 3, 3, 1, 1, 1 $ $4$ $3$ $( 2,11, 8)( 3,12, 6)( 4,10, 7)$
$ 6, 3, 2, 1 $ $12$ $6$ $( 2,12,11, 6, 8, 3)( 4, 7,10)( 5, 9)$
$ 2, 2, 2, 2, 2, 2 $ $9$ $2$ $( 1, 2)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)$
$ 3, 3, 3, 3 $ $8$ $3$ $( 1, 2, 3)( 4, 8,12)( 5, 6, 7)( 9,10,11)$
$ 6, 6 $ $6$ $6$ $( 1, 2, 9,10, 5, 6)( 3, 4,11,12, 7, 8)$
$ 3, 3, 3, 3 $ $8$ $3$ $( 1, 2,12)( 3, 7,11)( 4, 5, 6)( 8, 9,10)$
$ 2, 2, 2, 2, 2, 2 $ $3$ $2$ $( 1, 4)( 2,11)( 3, 6)( 5, 8)( 7,10)( 9,12)$
$ 3, 3, 3, 3 $ $2$ $3$ $( 1, 5, 9)( 2, 6,10)( 3, 7,11)( 4, 8,12)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $72=2^{3} \cdot 3^{2}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  72.44
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A 3B1 3B-1 3C1 3C-1 6A 6B1 6B-1
Size 1 3 3 9 2 4 4 8 8 6 12 12
2 P 1A 1A 1A 1A 3A 3B-1 3B1 3C-1 3C1 3A 3B1 3B-1
3 P 1A 2A 2B 2C 1A 1A 1A 1A 1A 2A 2B 2B
Type
72.44.1a R 1 1 1 1 1 1 1 1 1 1 1 1
72.44.1b R 1 1 −1 −1 1 1 1 1 1 1 −1 −1
72.44.1c1 C 1 1 1 1 1 ζ3−1 ζ3 ζ3−1 ζ3 1 ζ3 ζ3−1
72.44.1c2 C 1 1 1 1 1 ζ3 ζ3−1 ζ3 ζ3−1 1 ζ3−1 ζ3
72.44.1d1 C 1 1 −1 −1 1 ζ3−1 ζ3 ζ3−1 ζ3 1 −ζ3 −ζ3−1
72.44.1d2 C 1 1 −1 −1 1 ζ3 ζ3−1 ζ3 ζ3−1 1 −ζ3−1 −ζ3
72.44.2a R 2 2 0 0 −1 2 2 −1 −1 −1 0 0
72.44.2b1 C 2 2 0 0 −1 2ζ3−1 2ζ3 −ζ3−1 −ζ3 −1 0 0
72.44.2b2 C 2 2 0 0 −1 2ζ3 2ζ3−1 −ζ3 −ζ3−1 −1 0 0
72.44.3a R 3 −1 3 −1 3 0 0 0 0 −1 0 0
72.44.3b R 3 −1 −3 1 3 0 0 0 0 −1 0 0
72.44.6a R 6 −2 0 0 −3 0 0 0 0 1 0 0

magma: CharacterTable(G);