Properties

Label 12T43
Order \(72\)
n \(12\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $S_3\times A_4$

Related objects

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Group action invariants

Degree $n$ :  $12$
Transitive number $t$ :  $43$
Group :  $S_3\times A_4$
CHM label :  $A(4)[x]S(3)$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,7,10)(2,5,11)(3,6,9), (2,8,11)(3,6,12)(4,7,10), (1,5)(2,10)(4,8)(7,11), (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:  $S_3$, $C_6$
12:  $A_4$
18:  $S_3\times C_3$
24:  $A_4\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $S_3$

Degree 4: $A_4$

Degree 6: None

Low degree siblings

18T31, 18T32, 24T78, 24T83, 36T21, 36T50, 36T51

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

Group invariants

Order:  $72=2^{3} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [72, 44]
Character table:   
      2  3   1  3   1   1   1  3   .  2   .  3  2
      3  2   1  1   2   2   1  .   2  1   2  1  2

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

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

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