Group action invariants
| Degree $n$ : | $12$ | |
| Transitive number $t$ : | $250$ | |
| CHM label : | $[D(4)^{4}]S(3)=D(4)wrS(3)$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (3,6,9,12), (3,9), (1,5)(2,10)(4,8)(7,11), (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 7 4: $C_2^2$ x 7 6: $S_3$ 8: $C_2^3$ 12: $D_{6}$ x 3 24: $S_4$ x 3, $S_3 \times C_2^2$ 48: $S_4\times C_2$ x 9 96: $V_4^2:S_3$, 12T48 x 3 192: 12T100 x 3 384: 12T139 768: 16T1055 1536: 24T3386 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 3: $S_3$
Degree 4: None
Degree 6: $S_4\times C_2$
Low degree siblings
12T250 x 15, 24T5413 x 4, 24T5427 x 4, 24T5587 x 4, 24T5588 x 4, 24T5707 x 4, 24T5752 x 4, 24T6222 x 4, 24T6316 x 4, 24T6797 x 4, 24T7182 x 8, 24T7183 x 8, 24T7184 x 8, 24T7185 x 8, 24T7186 x 8, 24T7187 x 8, 24T7188 x 8, 24T7189 x 8, 24T7190 x 8, 24T7191 x 8, 24T7192 x 8, 24T7193 x 8, 24T7194 x 8, 24T7195 x 8, 24T7196 x 8Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy Classes
There are 65 conjugacy classes of elements. Data not shown.
Group invariants
| Order: | $3072=2^{10} \cdot 3$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | Data not available |
| Character table: Data not available. |