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Properties

Label 12T254
Order \(3456\)
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 $t$ :  $254$
CHM label :  $[1/3.A(4)^{3}]S(3)_{6}$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (3,6,12)(4,10,7), (3,12)(6,9), (3,9)(6,12), (1,5)(2,7)(4,8)(6,9)(10,11), (1,6)(3,7)(4,12)(8,11)(9,10)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
6:  $S_3$, $C_6$
18:  $S_3\times C_3$
54:  $(C_9:C_3):C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $S_3$

Degree 4: None

Degree 6: None

Low degree siblings

18T433, 18T434, 24T7219, 36T4273, 36T4541, 36T4567, 36T4569, 36T4571, 36T4575

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 $ $27$ $2$ $( 1, 7)( 3,12)( 4,10)( 6, 9)$
$ 2, 2, 2, 2, 2, 2 $ $27$ $2$ $( 1, 7)( 2, 5)( 3,12)( 4,10)( 6, 9)( 8,11)$
$ 3, 3, 1, 1, 1, 1, 1, 1 $ $48$ $3$ $( 4,10, 7)( 6, 9,12)$
$ 3, 3, 2, 2, 1, 1 $ $144$ $6$ $( 2, 5)( 4,10, 7)( 6, 9,12)( 8,11)$
$ 3, 3, 1, 1, 1, 1, 1, 1 $ $48$ $3$ $( 4, 7,10)( 6,12, 9)$
$ 3, 3, 2, 2, 1, 1 $ $144$ $6$ $( 2, 5)( 4, 7,10)( 6,12, 9)( 8,11)$
$ 3, 3, 3, 1, 1, 1 $ $128$ $3$ $( 4, 7,10)( 5,11, 8)( 6, 9,12)$
$ 9, 3 $ $384$ $9$ $( 1,12, 5, 4, 6,11,10, 9, 8)( 2, 7, 3)$
$ 9, 3 $ $384$ $9$ $( 1, 6,11, 7, 3, 2, 4, 9, 8)( 5,10,12)$
$ 9, 3 $ $384$ $9$ $( 1, 9, 8)( 2,10, 6,11, 4,12, 5, 7, 3)$
$ 2, 2, 2, 2, 2, 1, 1 $ $72$ $2$ $( 1, 5)( 2, 7)( 4, 8)( 6, 9)(10,11)$
$ 4, 2, 2, 2, 2 $ $72$ $4$ $( 1, 5)( 2, 7)( 3, 6,12, 9)( 4, 8)(10,11)$
$ 4, 4, 2, 1, 1 $ $216$ $4$ $( 1, 5, 7, 2)( 4, 8,10,11)( 6, 9)$
$ 4, 4, 4 $ $216$ $4$ $( 1, 5, 7, 2)( 3, 6,12, 9)( 4, 8,10,11)$
$ 6, 2, 2, 1, 1 $ $288$ $6$ $( 1, 5)( 2, 4, 8,10,11, 7)( 6,12)$
$ 6, 4, 2 $ $288$ $12$ $( 1, 5)( 2, 4, 8,10,11, 7)( 3,12, 9, 6)$
$ 6, 2, 2, 1, 1 $ $288$ $6$ $( 1, 5)( 2,10,11, 4, 8, 7)( 9,12)$
$ 6, 4, 2 $ $288$ $12$ $( 1, 5)( 2,10,11, 4, 8, 7)( 3,12, 6, 9)$

Group invariants

Order:  $3456=2^{7} \cdot 3^{3}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  Data not available
Character table:    
      2  7  7  7  7  3   3  3   3  .  .   .   .  4  4  4  4   2   2   2   2
      3  3  1  .  .  2   1  2   1  3  2   2   2  1  1  .  .   1   1   1   1

        1a 2a 2b 2c 3a  6a 3b  6b 3c 9a  9b  9c 2d 4a 4b 4c  6c 12a  6d 12b
     2P 1a 1a 1a 1a 3b  3b 3a  3a 3c 9a  9c  9b 1a 2a 2b 2c  3b  6b  3a  6a
     3P 1a 2a 2b 2c 1a  2a 1a  2a 1a 3c  3c  3c 2d 4a 4b 4c  2d  4a  2d  4a
     5P 1a 2a 2b 2c 3b  6b 3a  6a 3c 9a  9c  9b 2d 4a 4b 4c  6d 12b  6c 12a
     7P 1a 2a 2b 2c 3a  6a 3b  6b 3c 9a  9b  9c 2d 4a 4b 4c  6c 12a  6d 12b
    11P 1a 2a 2b 2c 3b  6b 3a  6a 3c 9a  9c  9b 2d 4a 4b 4c  6d 12b  6c 12a

X.1      1  1  1  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   1   1 -1 -1 -1 -1  -1  -1  -1  -1
X.3      1  1  1  1  A   A /A  /A  1  1   A  /A -1 -1 -1 -1  -A  -A -/A -/A
X.4      1  1  1  1 /A  /A  A   A  1  1  /A   A -1 -1 -1 -1 -/A -/A  -A  -A
X.5      1  1  1  1  A   A /A  /A  1  1   A  /A  1  1  1  1   A   A  /A  /A
X.6      1  1  1  1 /A  /A  A   A  1  1  /A   A  1  1  1  1  /A  /A   A   A
X.7      2  2  2  2  2   2  2   2  2 -1  -1  -1  .  .  .  .   .   .   .   .
X.8      2  2  2  2  B   B /B  /B  2 -1 -/A  -A  .  .  .  .   .   .   .   .
X.9      2  2  2  2 /B  /B  B   B  2 -1  -A -/A  .  .  .  .   .   .   .   .
X.10     6  6  6  6  .   .  .   . -3  .   .   .  .  .  .  .   .   .   .   .
X.11     9  5  1 -3  3  -1  3  -1  .  .   .   . -1  1 -1  1  -1   1  -1   1
X.12     9  5  1 -3  3  -1  3  -1  .  .   .   .  1 -1  1 -1   1  -1   1  -1
X.13     9  5  1 -3  C  -A /C -/A  .  .   .   . -1  1 -1  1  -A   A -/A  /A
X.14     9  5  1 -3 /C -/A  C  -A  .  .   .   . -1  1 -1  1 -/A  /A  -A   A
X.15     9  5  1 -3  C  -A /C -/A  .  .   .   .  1 -1  1 -1   A  -A  /A -/A
X.16     9  5  1 -3 /C -/A  C  -A  .  .   .   .  1 -1  1 -1  /A -/A   A  -A
X.17    27 -9  3 -1  .   .  .   .  .  .   .   . -3  3  1 -1   .   .   .   .
X.18    27 -9  3 -1  .   .  .   .  .  .   .   .  3 -3 -1  1   .   .   .   .
X.19    27  3 -5  3  .   .  .   .  .  .   .   . -3 -3  1  1   .   .   .   .
X.20    27  3 -5  3  .   .  .   .  .  .   .   .  3  3 -1 -1   .   .   .   .

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