Properties

Label 12T20
Order \(36\)
n \(12\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_3\times A_4$

Related objects

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Group action invariants

Degree $n$ :  $12$
Transitive number $t$ :  $20$
Group :  $C_3\times A_4$
CHM label :  $A(4)[x]C(3)$
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), (1,5,9)(2,6,10)(3,7,11)(4,8,12)
$|\Aut(F/K)|$:  $3$

Low degree resolvents

|G/N|Galois groups for stem field(s)
3:  $C_3$ x 4
9:  $C_3^2$
12:  $A_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $C_3$

Degree 4: $A_4$

Degree 6: None

Low degree siblings

12T20 x 2, 18T8, 36T12

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

Group invariants

Order:  $36=2^{2} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [36, 11]
Character table:   
      2  2  .  .  .   2  .  .   2  .  2  2  2
      3  2  2  2  2   1  2  2   1  2  1  2  2

        1a 3a 3b 3c  6a 3d 3e  6b 3f 2a 3g 3h
     2P 1a 3b 3a 3e  3h 3f 3c  3g 3d 1a 3h 3g
     3P 1a 1a 1a 1a  2a 1a 1a  2a 1a 2a 1a 1a
     5P 1a 3b 3a 3e  6b 3f 3c  6a 3d 2a 3h 3g

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

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