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) | |
| $|\Aut(F/K)|$: | $6$ | |
| 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 | |
| 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
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