Properties

Label 8T14
Degree $8$
Order $24$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $S_4$

Related objects

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Group action invariants

Degree $n$:  $8$
Transitive number $t$:  $14$
Group:  $S_4$
CHM label:  $S(4)[1/2]2=1/2(S_{4}[x]2)$
Parity:  $1$
Primitive:  no
Nilpotency class:  $-1$ (not nilpotent)
$|\Aut(F/K)|$:  $2$
Generators:  (1,3)(2,8)(4,6)(5,7), (1,2,3)(5,6,7), (1,4)(2,6)(3,7)(5,8)

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: $C_2$

Degree 4: $S_4$

Low degree siblings

4T5, 6T7, 6T8, 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 TypeSizeOrderRepresentative
$ 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 3, 3, 1, 1 $ $8$ $3$ $(2,3,8)(4,6,7)$
$ 2, 2, 2, 2 $ $3$ $2$ $(1,2)(3,8)(4,7)(5,6)$
$ 4, 4 $ $6$ $4$ $(1,4,3,6)(2,5,8,7)$
$ 2, 2, 2, 2 $ $6$ $2$ $(1,4)(2,6)(3,7)(5,8)$

Group invariants

Order:  $24=2^{3} \cdot 3$
Cyclic:  no
Abelian:  no
Solvable:  yes
GAP id:  [24, 12]
Character table:   
     2  3  .  3  2  2
     3  1  1  .  .  .

       1a 3a 2a 4a 2b
    2P 1a 3a 1a 2a 1a
    3P 1a 1a 2a 4a 2b

X.1     1  1  1  1  1
X.2     1  1  1 -1 -1
X.3     2 -1  2  .  .
X.4     3  . -1 -1  1
X.5     3  . -1  1 -1