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Properties

Label 12T166
Order \(576\)
n \(12\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No

Related objects

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Group action invariants

Degree $n$ :  $12$
Transitive number $t$ :  $166$
CHM label :  $[1/9.A(4)^{3}]3_{3}$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (3,12)(6,9), (3,9)(6,12), (2,8,11)(3,6,12)(4,7,10), (1,3,11,7,6,2,10,9,5)(4,12,8)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
3:  $C_3$
9:  $C_9$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $C_3$

Degree 4: None

Degree 6: None

Low degree siblings

12T166 x 6, 18T177 x 7, 36T954 x 7, 36T955 x 21, 36T956 x 7, 36T957 x 42

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 $ $9$ $2$ $( 3,12)( 6, 9)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $9$ $2$ $( 2, 5)( 3,12)( 6, 9)( 8,11)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $9$ $2$ $( 2, 5)( 3, 6)( 8,11)( 9,12)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $9$ $2$ $( 2, 5)( 3, 9)( 6,12)( 8,11)$
$ 2, 2, 2, 2, 2, 2 $ $9$ $2$ $( 1, 4)( 2, 5)( 3,12)( 6, 9)( 7,10)( 8,11)$
$ 2, 2, 2, 2, 2, 2 $ $9$ $2$ $( 1, 4)( 2, 5)( 3, 6)( 7,10)( 8,11)( 9,12)$
$ 2, 2, 2, 2, 2, 2 $ $9$ $2$ $( 1, 4)( 2,11)( 3, 9)( 5, 8)( 6,12)( 7,10)$
$ 3, 3, 3, 1, 1, 1 $ $64$ $3$ $( 4, 7,10)( 5,11, 8)( 6, 9,12)$
$ 3, 3, 3, 1, 1, 1 $ $64$ $3$ $( 4,10, 7)( 5, 8,11)( 6,12, 9)$
$ 9, 3 $ $64$ $9$ $( 1, 3,11, 7, 6, 2,10, 9, 5)( 4,12, 8)$
$ 9, 3 $ $64$ $9$ $( 1, 3, 8, 7, 9,11,10,12, 5)( 2, 4, 6)$
$ 9, 3 $ $64$ $9$ $( 1, 3, 5)( 2, 7,12,11, 4, 9, 8,10, 6)$
$ 9, 3 $ $64$ $9$ $( 1,11, 6,10, 5, 3, 7, 2, 9)( 4, 8,12)$
$ 9, 3 $ $64$ $9$ $( 1, 8, 6, 4, 5, 3,10,11, 9)( 2,12, 7)$
$ 9, 3 $ $64$ $9$ $( 1, 5, 3, 4,11,12,10, 8, 9)( 2, 6, 7)$

Group invariants

Order:  $576=2^{6} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [576, 8661]
Character table:    
      2  6  6  6  6  6  6  6  6  .  .  .  .  .  .  .  .
      3  2  .  .  .  .  .  .  .  2  2  2  2  2  2  2  2

        1a 2a 2b 2c 2d 2e 2f 2g 3a 3b 9a 9b 9c 9d 9e 9f
     2P 1a 1a 1a 1a 1a 1a 1a 1a 3b 3a 9d 9f 9e 9c 9b 9a
     3P 1a 2a 2b 2c 2d 2e 2f 2g 1a 1a 3b 3b 3b 3a 3a 3a
     5P 1a 2a 2b 2c 2d 2e 2f 2g 3b 3a 9f 9e 9d 9a 9c 9b
     7P 1a 2a 2b 2c 2d 2e 2f 2g 3a 3b 9b 9c 9a 9f 9d 9e

X.1      1  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  A  A  A /A /A /A
X.3      1  1  1  1  1  1  1  1  1  1 /A /A /A  A  A  A
X.4      1  1  1  1  1  1  1  1  A /A  B  C  D /C /B /D
X.5      1  1  1  1  1  1  1  1  A /A  C  D  B /D /C /B
X.6      1  1  1  1  1  1  1  1  A /A  D  B  C /B /D /C
X.7      1  1  1  1  1  1  1  1 /A  A /B /C /D  C  B  D
X.8      1  1  1  1  1  1  1  1 /A  A /D /B /C  B  D  C
X.9      1  1  1  1  1  1  1  1 /A  A /C /D /B  D  C  B
X.10     9  5  1  1  1 -3 -3 -3  .  .  .  .  .  .  .  .
X.11     9  1  1 -3 -3  5 -3  1  .  .  .  .  .  .  .  .
X.12     9 -3  1  5 -3  1  1 -3  .  .  .  .  .  .  .  .
X.13     9 -3  5 -3  1 -3  1  1  .  .  .  .  .  .  .  .
X.14     9 -3 -3  1  5  1 -3  1  .  .  .  .  .  .  .  .
X.15     9  1 -3 -3  1  1  5 -3  .  .  .  .  .  .  .  .
X.16     9  1 -3  1 -3 -3  1  5  .  .  .  .  .  .  .  .

A = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3
B = -E(9)^4-E(9)^7
C = E(9)^7
D = E(9)^4