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Properties

Label 8T13
Degree $8$
Order $24$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $A_4\times C_2$

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Show commands: Magma

magma: G := TransitiveGroup(8, 13);
 

Group action invariants

Degree $n$:  $8$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $13$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $A_4\times C_2$
CHM label:   $E(8):3=A(4)[x]2$
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)(2,8)(4,6)(5,7), (1,8)(2,3)(4,5)(6,7), (1,2,3)(4,6,5), (1,5)(2,6)(3,7)(4,8)
magma: Generators(G);
 

Low degree resolvents

$\card{(G/N)}$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: $C_2$

Degree 4: $A_4$

Low degree siblings

6T6, 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 TypeSizeOrderIndexRepresentative
1A $1^{8}$ $1$ $1$ $0$ $()$
2A $2^{4}$ $1$ $2$ $4$ $(1,6)(2,5)(3,4)(7,8)$
2B $2^{4}$ $3$ $2$ $4$ $(1,3)(2,8)(4,6)(5,7)$
2C $2^{4}$ $3$ $2$ $4$ $(1,4)(2,7)(3,6)(5,8)$
3A1 $3^{2},1^{2}$ $4$ $3$ $4$ $(1,2,3)(4,6,5)$
3A-1 $3^{2},1^{2}$ $4$ $3$ $4$ $(1,3,2)(4,5,6)$
6A1 $6,2$ $4$ $6$ $6$ $(1,4,2,6,3,5)(7,8)$
6A-1 $6,2$ $4$ $6$ $6$ $(1,5,3,6,2,4)(7,8)$

Malle's constant $a(G)$:     $1/4$

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 ζ3−1 ζ3 ζ3 ζ3−1
24.13.1c2 C 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3
24.13.1d1 C 1 −1 1 −1 ζ3−1 ζ3 −ζ3 −ζ3−1
24.13.1d2 C 1 −1 1 −1 ζ3 ζ3−1 −ζ3−1 −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);