Properties

Label 12T32
Order \(48\)
n \(12\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2^4:C_3$

Related objects

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Group action invariants

Degree $n$ :  $12$
Transitive number $t$ :  $32$
Group :  $C_2^4:C_3$
CHM label :  $[E(4)^{2}]3$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,10)(3,12)(4,7)(6,9), (1,7)(3,9)(4,10)(6,12), (1,5,9)(2,6,10)(3,7,11)(4,8,12)
$|\Aut(F/K)|$:  $4$

Low degree resolvents

|G/N|Galois groups for stem field(s)
3:  $C_3$
12:  $A_4$ x 5

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $C_3$

Degree 4: None

Degree 6: $A_4$ x 3

Low degree siblings

12T32 x 9, 16T64, 24T59 x 5

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

Group invariants

Order:  $48=2^{4} \cdot 3$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [48, 50]
Character table:   
     2  4  4  4  4  .  .  4  4
     3  1  .  .  .  1  1  .  .

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

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

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