↑

Properties

Label 8T33
Degree $8$
Order $96$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2^4:C_6$

Related objects

Downloads

Learn more

Show commands: Magma

magma: G := TransitiveGroup(8, 33);
 

Group action invariants

Degree $n$:  $8$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $33$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_2^4:C_6$
CHM label:   $E(8):A_{4}=[1/3.A(4)^{2}]2=E(4):6$
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:  (1,3)(2,8)(4,6)(5,7), (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

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: None

Low degree siblings

8T33, 12T58 x 2, 12T59 x 2, 16T183, 24T181 x 2, 24T182 x 2, 24T183 x 2, 24T184 x 2, 24T185, 24T186, 32T389

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 $ $1$ $1$ $()$
$ 2, 2, 1, 1, 1, 1 $ $6$ $2$ $(4,5)(6,7)$
$ 3, 3, 1, 1 $ $16$ $3$ $(2,3,8)(5,7,6)$
$ 3, 3, 1, 1 $ $16$ $3$ $(2,8,3)(5,6,7)$
$ 2, 2, 2, 2 $ $6$ $2$ $(1,2)(3,8)(4,5)(6,7)$
$ 2, 2, 2, 2 $ $3$ $2$ $(1,2)(3,8)(4,7)(5,6)$
$ 6, 2 $ $16$ $6$ $(1,4)(2,5,3,7,8,6)$
$ 6, 2 $ $16$ $6$ $(1,4)(2,6,8,7,3,5)$
$ 2, 2, 2, 2 $ $4$ $2$ $(1,4)(2,7)(3,6)(5,8)$
$ 4, 4 $ $12$ $4$ $(1,4,2,7)(3,6,8,5)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $96=2^{5} \cdot 3$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  96.70
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 2D 3A1 3A-1 4A 6A1 6A-1
Size 1 3 4 6 6 16 16 12 16 16
2 P 1A 1A 1A 1A 1A 3A-1 3A1 2A 3A1 3A-1
3 P 1A 2A 2B 2C 2D 1A 1A 4A 2B 2B
Type
96.70.1a R 1 1 1 1 1 1 1 1 1 1
96.70.1b R 1 1 −1 1 1 1 1 −1 −1 −1
96.70.1c1 C 1 1 1 1 1 ζ3−1 ζ3 1 ζ3 ζ3−1
96.70.1c2 C 1 1 1 1 1 ζ3 ζ3−1 1 ζ3−1 ζ3
96.70.1d1 C 1 1 −1 1 1 ζ3−1 ζ3 −1 −ζ3 −ζ3−1
96.70.1d2 C 1 1 −1 1 1 ζ3 ζ3−1 −1 −ζ3−1 −ζ3
96.70.3a R 3 3 3 −1 −1 0 0 −1 0 0
96.70.3b R 3 3 −3 −1 −1 0 0 1 0 0
96.70.6a R 6 −2 0 −2 2 0 0 0 0 0
96.70.6b R 6 −2 0 2 −2 0 0 0 0 0

magma: CharacterTable(G);