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Properties

Label 6T7
Order \(24\)
n \(6\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $S_4$

Related objects

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Group action invariants

Degree $n$ :  $6$
Transitive number $t$ :  $7$
Group :  $S_4$
CHM label :  $S_{4}(6d) = [2^{2}]S(3)$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,4)(2,5), (1,5)(2,4), (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, 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 TypeSizeOrderRepresentative
$ 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 2, 2, 1, 1 $ $6$ $2$ $(2,3)(5,6)$
$ 2, 2, 1, 1 $ $3$ $2$ $(2,5)(3,6)$
$ 3, 3 $ $8$ $3$ $(1,2,3)(4,5,6)$
$ 4, 2 $ $6$ $4$ $(1,2,4,5)(3,6)$

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

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