# Properties

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

# Learn more

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

 Label Cycle Type Size Order Representative 1A $1^{8}$ $1$ $1$ $()$ 2A $2^{4}$ $1$ $2$ $(1,6)(2,5)(3,4)(7,8)$ 2B $2^{4}$ $3$ $2$ $(1,7)(2,4)(3,5)(6,8)$ 2C $2^{4}$ $3$ $2$ $(1,8)(2,3)(4,5)(6,7)$ 2D $2^{2},1^{4}$ $6$ $2$ $(1,3)(4,6)$ 2E $2^{4}$ $6$ $2$ $(1,6)(2,4)(3,5)(7,8)$ 3A $3^{2},1^{2}$ $8$ $3$ $(1,2,3)(4,6,5)$ 4A $4^{2}$ $6$ $4$ $(1,8,3,2)(4,5,6,7)$ 4B $4^{2}$ $6$ $4$ $(1,7,2,4)(3,6,8,5)$ 6A $6,2$ $8$ $6$ $(1,4,2,6,3,5)(7,8)$

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 2C 2C 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);