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Properties

Label 8T24
Degree $8$
Order $48$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $S_4\times C_2$

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Show commands: Magma

magma: G := TransitiveGroup(8, 24);
 

Group action invariants

Degree $n$:  $8$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $24$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $S_4\times C_2$
CHM label:   $E(8):D_{6}=S(4)[x]2$
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), (2,3)(4,5), (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)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$6$:  $S_3$
$12$:  $D_{6}$
$24$:  $S_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $S_4$

Low degree siblings

6T11 x 2, 8T24, 12T21, 12T22, 12T23 x 2, 12T24 x 2, 16T61, 24T46, 24T47, 24T48 x 2

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $48=2^{4} \cdot 3$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  48.48
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 2D 2E 3A 4A 4B 6A
Size 1 1 3 3 6 6 8 6 6 8
2 P 1A 1A 1A 1A 1A 1A 3A 2B 2B 3A
3 P 1A 2A 2B 2C 2D 2E 1A 4A 4B 2A
Type
48.48.1a R 1 1 1 1 1 1 1 1 1 1
48.48.1b R 1 −1 1 −1 −1 1 1 −1 1 −1
48.48.1c R 1 −1 1 −1 1 −1 1 1 −1 −1
48.48.1d R 1 1 1 1 −1 −1 1 −1 −1 1
48.48.2a R 2 2 2 2 0 0 −1 0 0 −1
48.48.2b R 2 −2 2 −2 0 0 −1 0 0 1
48.48.3a R 3 3 −1 −1 1 1 0 −1 −1 0
48.48.3b R 3 3 −1 −1 −1 −1 0 1 1 0
48.48.3c R 3 −3 −1 1 −1 1 0 1 −1 0
48.48.3d R 3 −3 −1 1 1 −1 0 −1 1 0

magma: CharacterTable(G);