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) | |
$\card{\Aut(F/K)}$: | $6$ | |
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) |
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, 36T6Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
Cycle Type | Size | Order | Representative |
$ 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 | |
Label: | 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 |