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Properties

Label 12T9
Order \(24\)
n \(12\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $S_4$

Related objects

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Group action invariants

Degree $n$ :  $12$
Transitive number $t$ :  $9$
Group :  $S_4$
CHM label :  $1/2[1/8.2^{6}]S(3)=S_{4}(12e)$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,2)(3,12)(4,11)(5,10)(6,9)(7,8), (1,7)(3,9)(4,10)(6,12), (1,5,9)(2,6,10)(3,7,11)(4,8,12)
$|\Aut(F/K)|$:  $4$

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 3: $S_3$

Degree 4: None

Degree 6: $S_3$, $S_4$, $S_4$

Low degree siblings

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

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