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Properties

Label 12T8
Order \(24\)
n \(12\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $S_4$

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Group action invariants

Degree $n$ :  $12$
Transitive number $t$ :  $8$
Group :  $S_4$
CHM label :  $S_{4}(12d)$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,2)(3,5)(4,6)(7,9)(8,10), (1,3,6,12)(2,4,7,10)(5,8,11,9)
$|\Aut(F/K)|$:  $2$

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: $S_3$

Degree 4: $S_4$

Degree 6: $S_4$

Low degree siblings

4T5, 6T7, 6T8, 8T14, 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, 1, 1 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 2, 1, 1 $ $6$ $2$ $( 2,11)( 3, 4)( 5,10)( 6, 8)( 7,12)$
$ 3, 3, 3, 3 $ $8$ $3$ $( 1, 2,11)( 3, 6,10)( 4, 5, 8)( 7,12, 9)$
$ 4, 4, 4 $ $6$ $4$ $( 1, 3, 6,12)( 2, 4, 7,10)( 5, 8,11, 9)$
$ 2, 2, 2, 2, 2, 2 $ $3$ $2$ $( 1, 6)( 2, 7)( 3,12)( 4,10)( 5,11)( 8, 9)$

Group invariants

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

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