Group action invariants
| Degree $n$ : | $12$ | |
| Transitive number $t$ : | $274$ | |
| CHM label : | $[S(3)^{4}]D(4)=S(3)wrD(4)$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,7)(3,9)(5,11), (4,8), (4,8,12), (1,4,7,10)(2,5,8,11)(3,6,9,12) | |
| $|\Aut(F/K)|$: | $1$ |
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: $C_2^2 \wr C_2$ 64: $(((C_4 \times C_2): C_2):C_2):C_2$ 128: $C_2 \wr C_2\wr C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: None
Degree 4: $D_{4}$
Degree 6: None
Low degree siblings
12T274, 18T555, 24T10135 x 2, 24T10136 x 2, 24T10137 x 2, 24T10138 x 2, 24T10139 x 2, 24T10140 x 2, 24T10141 x 2, 24T10142 x 2, 24T10143 x 2, 24T10144 x 2, 24T10145 x 2, 24T10146 x 2, 24T10147 x 2, 24T10148 x 2, 24T10149 x 2, 24T10165 x 2, 36T8977, 36T8978, 36T8979, 36T8980 x 2, 36T8981 x 2, 36T8982 x 2, 36T8983 x 2, 36T8984 x 2, 36T8985 x 2, 36T9087 x 2Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy Classes
There are 54 conjugacy classes of elements. Data not shown.
Group invariants
| Order: | $10368=2^{7} \cdot 3^{4}$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | Data not available |
| Character table: Data not available. |