# Properties

 Label 4T5 Degree $4$ Order $24$ Cyclic no Abelian no Solvable yes Primitive yes $p$-group no Group: $S_4$

# Related objects

Show commands: Magma

magma: G := TransitiveGroup(4, 5);

## Group action invariants

 Degree $n$: $4$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $5$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $S_4$ CHM label: $S4$ Parity: $-1$ magma: IsEven(G); Primitive: yes magma: IsPrimitive(G); Nilpotency class: $-1$ (not nilpotent) magma: NilpotencyClass(G); $\card{\Aut(F/K)}$: $1$ magma: Order(Centralizer(SymmetricGroup(n), G)); Generators: (1,2,3,4), (1,2) magma: Generators(G);

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$6$:  $S_3$

Resolvents shown for degrees $\leq 47$

Degree 2: None

## Low degree siblings

6T7, 6T8, 8T14, 12T8, 12T9, 24T10

Siblings are shown with degree $\leq 47$

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

## Conjugacy classes

 Cycle Type Size Order Representative $1, 1, 1, 1$ $1$ $1$ $()$ $2, 1, 1$ $6$ $2$ $(3,4)$ $3, 1$ $8$ $3$ $(2,3,4)$ $2, 2$ $3$ $2$ $(1,2)(3,4)$ $4$ $6$ $4$ $(1,2,3,4)$

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); Label: 24.12 magma: IdentifyGroup(G);
 Character table:  2 3 2 . 3 2 3 1 . 1 . . 1a 2a 3a 2b 4a 2P 1a 1a 3a 1a 2b 3P 1a 2a 1a 2b 4a X.1 1 -1 1 1 -1 X.2 3 -1 . -1 1 X.3 2 . -1 2 . X.4 3 1 . -1 -1 X.5 1 1 1 1 1 

magma: CharacterTable(G);