Properties

Label 12T126
Order \(288\)
n \(12\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $A_4\wr C_2$

Related objects

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Group action invariants

Degree $n$ :  $12$
Transitive number $t$ :  $126$
Group :  $A_4\wr C_2$
CHM label :  $[A_{4}^{2}]2=A_{4}wr2$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (2,8)(4,10), (2,6,10)(4,8,12), (1,12)(2,3)(4,5)(6,7)(8,9)(10,11)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
6:  $S_3$, $C_6$
18:  $S_3\times C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 4: None

Degree 6: $S_3\times C_3$

Low degree siblings

8T42, 12T128, 12T129, 16T708, 18T112, 18T113, 24T692, 24T694, 24T695, 24T702, 24T703, 24T704, 32T9306, 36T316, 36T318, 36T456, 36T457, 36T458, 36T459

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, 1, 1, 1, 1, 1, 1, 1, 1 $ $6$ $2$ $( 4,10)( 6,12)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $9$ $2$ $( 3, 9)( 4,10)( 5,11)( 6,12)$
$ 3, 3, 1, 1, 1, 1, 1, 1 $ $8$ $3$ $( 2, 4, 6)( 8,10,12)$
$ 3, 3, 2, 2, 1, 1 $ $24$ $6$ $( 2, 4, 6)( 3, 9)( 5,11)( 8,10,12)$
$ 3, 3, 1, 1, 1, 1, 1, 1 $ $8$ $3$ $( 2, 6, 4)( 8,12,10)$
$ 3, 3, 2, 2, 1, 1 $ $24$ $6$ $( 2, 6, 4)( 3, 9)( 5,11)( 8,12,10)$
$ 2, 2, 2, 2, 2, 2 $ $12$ $2$ $( 1, 2)( 3, 4)( 5,12)( 6,11)( 7, 8)( 9,10)$
$ 4, 4, 2, 2 $ $36$ $4$ $( 1, 2)( 3, 4, 9,10)( 5,12,11, 6)( 7, 8)$
$ 6, 6 $ $48$ $6$ $( 1, 2, 3, 4, 5,12)( 6, 7, 8, 9,10,11)$
$ 6, 6 $ $48$ $6$ $( 1, 2, 5,12, 3, 4)( 6, 9,10, 7, 8,11)$
$ 3, 3, 3, 3 $ $16$ $3$ $( 1, 3, 5)( 2, 4, 6)( 7, 9,11)( 8,10,12)$
$ 3, 3, 3, 3 $ $32$ $3$ $( 1, 3, 5)( 2, 6, 4)( 7, 9,11)( 8,12,10)$
$ 3, 3, 3, 3 $ $16$ $3$ $( 1, 5, 3)( 2, 6, 4)( 7,11, 9)( 8,12,10)$

Group invariants

Order:  $288=2^{5} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [288, 1025]
Character table:   
      2  5  4  5   2   2   2   2  3  3   1   1  1  .  1
      3  2  1  .   2   1   2   1  1  .   1   1  2  2  2

        1a 2a 2b  3a  6a  3b  6b 2c 4a  6c  6d 3c 3d 3e
     2P 1a 1a 1a  3b  3b  3a  3a 1a 2b  3c  3e 3e 3d 3c
     3P 1a 2a 2b  1a  2a  1a  2a 2c 4a  2c  2c 1a 1a 1a
     5P 1a 2a 2b  3b  6b  3a  6a 2c 4a  6d  6c 3e 3d 3c

X.1      1  1  1   1   1   1   1  1  1   1   1  1  1  1
X.2      1  1  1   1   1   1   1 -1 -1  -1  -1  1  1  1
X.3      1  1  1   A   A  /A  /A -1 -1  -A -/A /A  1  A
X.4      1  1  1  /A  /A   A   A -1 -1 -/A  -A  A  1 /A
X.5      1  1  1   A   A  /A  /A  1  1   A  /A /A  1  A
X.6      1  1  1  /A  /A   A   A  1  1  /A   A  A  1 /A
X.7      2  2  2  -1  -1  -1  -1  .  .   .   .  2 -1  2
X.8      2  2  2  -A  -A -/A -/A  .  .   .   .  C -1 /C
X.9      2  2  2 -/A -/A  -A  -A  .  .   .   . /C -1  C
X.10     6  2 -2   3  -1   3  -1  .  .   .   .  .  .  .
X.11     6  2 -2   B  -A  /B -/A  .  .   .   .  .  .  .
X.12     6  2 -2  /B -/A   B  -A  .  .   .   .  .  .  .
X.13     9 -3  1   .   .   .   . -3  1   .   .  .  .  .
X.14     9 -3  1   .   .   .   .  3 -1   .   .  .  .  .

A = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3
B = 3*E(3)^2
  = (-3-3*Sqrt(-3))/2 = -3-3b3
C = 2*E(3)
  = -1+Sqrt(-3) = 2b3