Group action invariants
| Degree $n$: | $12$ | |
| Transitive number $t$: | $5$ | |
| Group: | $C_3 : C_4$ | |
| CHM label: | $1/2[3:2]4$ | |
| Parity: | $-1$ | |
| Primitive: | no | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| $|\Aut(F/K)|$: | $12$ | |
| Generators: | (1,8,7,2)(3,6,9,12)(4,11,10,5), (1,5,9)(2,6,10)(3,7,11)(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) $2$: $C_2$ $4$: $C_4$ $6$: $S_3$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: $S_3$
Degree 4: $C_4$
Degree 6: $S_3$
Low degree siblings
There are no siblings 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 $ | $1$ | $1$ | $()$ |
| $ 4, 4, 4 $ | $3$ | $4$ | $( 1, 2, 7, 8)( 3,12, 9, 6)( 4, 5,10,11)$ |
| $ 6, 6 $ | $2$ | $6$ | $( 1, 3, 5, 7, 9,11)( 2, 4, 6, 8,10,12)$ |
| $ 4, 4, 4 $ | $3$ | $4$ | $( 1, 4, 7,10)( 2, 9, 8, 3)( 5,12,11, 6)$ |
| $ 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 5, 9)( 2, 6,10)( 3, 7,11)( 4, 8,12)$ |
| $ 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12)$ |
Group invariants
| Order: | $12=2^{2} \cdot 3$ | |
| Cyclic: | no | |
| Abelian: | no | |
| Solvable: | yes | |
| GAP id: | [12, 1] |
| Character table: |
2 2 2 1 2 1 2
3 1 . 1 . 1 1
1a 4a 6a 4b 3a 2a
2P 1a 2a 3a 2a 3a 1a
3P 1a 4b 2a 4a 1a 2a
5P 1a 4a 6a 4b 3a 2a
X.1 1 1 1 1 1 1
X.2 1 -1 1 -1 1 1
X.3 1 A -1 -A 1 -1
X.4 1 -A -1 A 1 -1
X.5 2 . 1 . -1 -2
X.6 2 . -1 . -1 2
A = -E(4)
= -Sqrt(-1) = -i
|