Group action invariants
| Degree $n$ : | $12$ | |
| Transitive number $t$ : | $260$ | |
| CHM label : | $[2S_{4}^{2}]2=2S_{4}wr2$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (6,12), (2,6,10)(4,8,12), (2,10)(4,8), (1,12)(2,3)(4,5)(6,7)(8,9)(10,11) | |
| $|\Aut(F/K)|$: | $2$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ x 7 4: $C_2^2$ x 7 8: $D_{4}$ x 6, $C_2^3$ 16: $D_4\times C_2$ x 3 32: $C_2^2 \wr C_2$ 72: $C_3^2:D_4$ 144: 12T77 288: 12T125 1152: $S_4\wr C_2$ 2304: 12T235 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: None
Degree 4: None
Degree 6: $C_3^2:D_4$
Low degree siblings
12T260 x 7, 16T1648 x 8, 24T7465 x 4, 24T7517 x 4, 24T7518 x 4, 24T7519 x 4, 24T7520 x 8, 24T7521 x 8, 24T7522 x 8, 24T7523 x 8, 24T7524 x 8, 24T7525 x 8, 24T7526 x 8, 24T7527 x 8, 24T7528 x 4, 24T7529 x 8, 24T7530 x 4, 32T396906 x 4, 32T396907 x 8, 32T396908 x 4, 32T396909 x 4, 32T396910 x 4, 32T396911 x 4, 36T5190 x 4, 36T5191 x 8, 36T5271 x 4, 36T5540 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: | $4608=2^{9} \cdot 3^{2}$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | Data not available |
| Character table: Data not available. |