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Properties

Label 12T18
Order \(36\)
n \(12\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_6\times S_3$

Related objects

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Group action invariants

Degree $n$ :  $12$
Transitive number $t$ :  $18$
Group :  $C_6\times S_3$
CHM label :  $[3^{2}]E(4)$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,10)(2,5)(3,12)(4,7)(6,9)(8,11), (2,6,10)(4,8,12), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)
$|\Aut(F/K)|$:  $6$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
3:  $C_3$
4:  $C_2^2$
6:  $S_3$, $C_6$ x 3
12:  $D_{6}$, $C_6\times C_2$
18:  $S_3\times C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 3: None

Degree 4: $C_2^2$

Degree 6: $S_3\times C_3$

Low degree siblings

18T6 x 2, 36T6

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

Group invariants

Order:  $36=2^{2} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [36, 12]
Character table:    
      2  2   1   1  2   2   2   2   1  1   2   2  2   2  1  2   1   2   2
      3  2   2   2  1   1   1   2   2  2   1   1  1   2  2  2   2   2   2

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

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

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