Group action invariants
| Degree $n$ : | $12$ | |
| Transitive number $t$ : | $222$ | |
| Group : | $D_4\wr C_3$ | |
| CHM label : | $[D(4)^{4}]3=D(4)wr3$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (3,6,9,12), (3,9), (1,5,9)(2,6,10)(3,7,11)(4,8,12) | |
| $|\Aut(F/K)|$: | $2$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ x 3 3: $C_3$ 4: $C_2^2$ 6: $C_6$ x 3 12: $A_4$ x 5, $C_6\times C_2$ 24: $A_4\times C_2$ x 15 48: $C_2^2 \times A_4$ x 5, $C_2^4:C_3$ 96: 12T56 x 3 192: 12T90 384: 16T718 768: 24T1654 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 3: $C_3$
Degree 4: None
Degree 6: $A_4\times C_2$
Low degree siblings
12T222 x 31, 24T4838 x 32, 24T4839 x 16, 24T4840 x 32, 24T4841 x 16, 24T4842 x 16, 24T4843 x 32, 24T4844 x 32, 24T4845 x 32, 24T4846 x 32, 24T4847 x 32, 24T4848 x 32, 24T4849 x 32, 24T4850 x 32, 24T4851 x 16, 24T4852 x 32, 24T4853 x 16, 24T4854 x 32, 24T4855 x 16, 24T4856 x 16Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy Classes
There are 55 conjugacy classes of elements. Data not shown.
Group invariants
| Order: | $1536=2^{9} \cdot 3$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | Data not available |
| Character table: Data not available. |