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Properties

Label 12T129
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 $t$ :  $129$
Group :  $A_4\wr C_2$
CHM label :  $[1/4E(4)^{3}:3:2]3$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,10)(3,12)(4,7)(6,9), (1,7,10)(2,5,11)(3,6,9), (4,7)(6,9)(8,11), (1,5,9)(2,6,10)(3,7,11)(4,8,12)
$|\Aut(F/K)|$:  $1$

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: None

Degree 3: $C_3$

Degree 4: None

Degree 6: None

Low degree siblings

8T42, 12T126, 12T128, 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, 2, 1, 1, 1, 1, 1, 1 $ $12$ $2$ $( 5, 8)( 7,10)( 9,12)$
$ 3, 3, 3, 1, 1, 1 $ $32$ $3$ $( 4, 7,10)( 5,11, 8)( 6, 9,12)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $9$ $2$ $( 2, 5)( 3,12)( 6, 9)( 8,11)$
$ 4, 4, 2, 1, 1 $ $36$ $4$ $( 2, 5,11, 8)( 3,12, 6, 9)( 7,10)$
$ 6, 3, 3 $ $48$ $6$ $( 1, 2, 3)( 4, 5, 6,10,11,12)( 7, 8, 9)$
$ 3, 3, 3, 3 $ $8$ $3$ $( 1, 2, 3)( 4, 5, 9)( 6,10, 8)( 7,11,12)$
$ 3, 3, 3, 3 $ $16$ $3$ $( 1, 2, 3)( 4,11, 6)( 5,12,10)( 7, 8, 9)$
$ 6, 6 $ $24$ $6$ $( 1, 2, 6, 7,11, 9)( 3, 4, 5,12,10, 8)$
$ 6, 3, 3 $ $48$ $6$ $( 1, 3, 2)( 4, 6,11)( 5, 7,12, 8,10, 9)$
$ 3, 3, 3, 3 $ $16$ $3$ $( 1, 3, 2)( 4, 6,11)( 5,10,12)( 7, 9, 8)$
$ 3, 3, 3, 3 $ $8$ $3$ $( 1, 3, 2)( 4, 9, 5)( 6, 8,10)( 7,12,11)$
$ 6, 6 $ $24$ $6$ $( 1, 3,11,10, 6, 5)( 2, 4, 9, 8, 7,12)$
$ 2, 2, 2, 2, 2, 2 $ $6$ $2$ $( 1, 4)( 2, 5)( 3, 9)( 6,12)( 7,10)( 8,11)$

Group invariants

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

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

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

A = -E(3)
  = (1-Sqrt(-3))/2 = -b3
B = 3*E(3)^2
  = (-3-3*Sqrt(-3))/2 = -3-3b3
C = 2*E(3)^2
  = -1-Sqrt(-3) = -1-i3