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Properties

Label 12T6
Order \(24\)
n \(12\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $A_4\times C_2$

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Group action invariants

Degree $n$ :  $12$
Transitive number $t$ :  $6$
Group :  $A_4\times C_2$
CHM label :  $A_{4}(12)x2$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,9,5)(2,4,3)(6,8,7)(10,12,11), (1,7)(2,11)(3,12)(4,10)(5,8)(6,9), (1,11,6)(2,9,7)(3,10,5)(4,8,12)
$|\Aut(F/K)|$:  $4$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
6:  $C_6$
12:  $A_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $C_3$

Degree 4: None

Degree 6: $A_4$, $A_4\times C_2$

Low degree siblings

6T6, 8T13, 12T7, 24T9

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

Group invariants

Order:  $24=2^{3} \cdot 3$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [24, 13]
Character table:    
     2  3  3   1   1   1   1  3  3
     3  1  .   1   1   1   1  .  1

       1a 2a  6a  3a  6b  3b 2b 2c
    2P 1a 1a  3b  3b  3a  3a 1a 1a
    3P 1a 2a  2c  1a  2c  1a 2b 2c
    5P 1a 2a  6b  3b  6a  3a 2b 2c

X.1     1  1   1   1   1   1  1  1
X.2     1 -1  -1   1  -1   1  1 -1
X.3     1 -1   A  -A  /A -/A  1 -1
X.4     1 -1  /A -/A   A  -A  1 -1
X.5     1  1 -/A -/A  -A  -A  1  1
X.6     1  1  -A  -A -/A -/A  1  1
X.7     3  1   .   .   .   . -1 -3
X.8     3 -1   .   .   .   . -1  3

A = -E(3)
  = (1-Sqrt(-3))/2 = -b3