Group action invariants
| Degree $n$ : | $12$ | |
| Transitive number $t$ : | $167$ | |
| CHM label : | $[3^{4}]D(4)=3wrD(4)$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,7)(3,9)(5,11), (4,8,12), (1,4,7,10)(2,5,8,11)(3,6,9,12) | |
| $|\Aut(F/K)|$: | $3$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ x 3 3: $C_3$ 4: $C_2^2$ 6: $S_3$, $C_6$ x 3 8: $D_{4}$ 12: $D_{6}$, $C_6\times C_2$ 18: $S_3\times C_3$ 24: $(C_6\times C_2):C_2$, $D_4 \times C_3$ 36: $C_6\times S_3$ 72: $C_3^2:D_4$, 12T42 216: 12T116, 12T121 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: None
Degree 4: $D_{4}$
Degree 6: None
Low degree siblings
12T167, 18T189 x 2, 24T1519 x 2, 24T1536, 36T1079 x 2, 36T1080 x 2, 36T1081 x 2, 36T1158, 36T1163, 36T1165, 36T1170, 36T1180, 36T1191 x 2, 36T1200 x 2, 36T1231 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: | $648=2^{3} \cdot 3^{4}$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | [648, 719] |
| Character table: Data not available. |