Properties

 Label 6T8 Order $$24$$ n $$6$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No Group: $S_4$

Related objects

Group action invariants

 Degree $n$ : $6$ Transitive number $t$ : $8$ Group : $S_4$ CHM label : $S_{4}(6c) = 1/2[2^{3}]S(3)$ Parity: $-1$ Primitive: No Nilpotency class: $-1$ (not nilpotent) Generators: (1,5)(2,4)(3,6), (1,4)(2,5), (1,3,5)(2,4,6) $|\Aut(F/K)|$: $2$

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $S_3$

Low degree siblings

4T5, 6T7, 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 $1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $4, 1, 1$ $6$ $4$ $(2,3,5,6)$ $2, 2, 1, 1$ $3$ $2$ $(2,5)(3,6)$ $2, 2, 2$ $6$ $2$ $(1,2)(3,6)(4,5)$ $3, 3$ $8$ $3$ $(1,2,3)(4,5,6)$

Group invariants

 Order: $24=2^{3} \cdot 3$ Cyclic: No Abelian: No Solvable: Yes GAP id: [24, 12]
 Character table:  2 3 2 3 2 . 3 1 . . . 1 1a 4a 2a 2b 3a 2P 1a 2a 1a 1a 3a 3P 1a 4a 2a 2b 1a X.1 1 1 1 1 1 X.2 1 -1 1 -1 1 X.3 2 . 2 . -1 X.4 3 -1 -1 1 . X.5 3 1 -1 -1 .