Properties

Label 6T6
Degree $6$
Order $24$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $A_4\times C_2$

Related objects

Downloads

Learn more

Show commands: Magma

magma: G := TransitiveGroup(6, 6);
 

Group action invariants

Degree $n$:  $6$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $6$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $A_4\times C_2$
CHM label:   $2A_{4}(6) = [2^{3}]3 = 2 wr 3$
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:  (1,3,5)(2,4,6), (3,6)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $C_6$
$12$:  $A_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $C_3$

Low degree siblings

8T13, 12T6, 12T7, 24T9

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderRepresentative
$1^{6}$ $1$ $1$ $()$
$2,1^{4}$ $3$ $2$ $(3,6)$
$2^{2},1^{2}$ $3$ $2$ $(2,5)(3,6)$
$3^{2}$ $4$ $3$ $(1,2,3)(4,5,6)$
$6$ $4$ $6$ $(1,2,3,4,5,6)$
$3^{2}$ $4$ $3$ $(1,3,2)(4,6,5)$
$6$ $4$ $6$ $(1,3,5,4,6,2)$
$2^{3}$ $1$ $2$ $(1,4)(2,5)(3,6)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $24=2^{3} \cdot 3$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  24.13
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A1 3A-1 6A1 6A-1
Size 1 1 3 3 4 4 4 4
2 P 1A 1A 1A 1A 3A-1 3A1 3A1 3A-1
3 P 1A 2A 2B 2C 1A 1A 2A 2A
Type
24.13.1a R 1 1 1 1 1 1 1 1
24.13.1b R 1 1 1 1 1 1 1 1
24.13.1c1 C 1 1 1 1 ζ31 ζ3 ζ3 ζ31
24.13.1c2 C 1 1 1 1 ζ3 ζ31 ζ31 ζ3
24.13.1d1 C 1 1 1 1 ζ31 ζ3 ζ3 ζ31
24.13.1d2 C 1 1 1 1 ζ3 ζ31 ζ31 ζ3
24.13.3a R 3 3 1 1 0 0 0 0
24.13.3b R 3 3 1 1 0 0 0 0

magma: CharacterTable(G);