Group action invariants
| Degree $n$ : | $12$ | |
| Transitive number $t$ : | $206$ | |
| CHM label : | $[1/9.A(4)^{3}]S(3)$ | |
| Parity: | $1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (3,12)(6,9), (3,9)(6,12), (2,8,11)(3,6,12)(4,7,10), (1,5)(2,10)(4,8)(7,11), (1,5,9)(2,6,10)(3,7,11)(4,8,12) | |
| $|\Aut(F/K)|$: | $1$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ 3: $C_3$ 6: $S_3$, $C_6$ 12: $A_4$ 18: $S_3\times C_3$ 24: $A_4\times C_2$ 72: 12T43 288: $A_4\wr C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 3: $S_3$
Degree 4: None
Degree 6: None
Low degree siblings
12T206 x 2, 18T268 x 3, 18T271 x 3, 24T2768 x 3, 24T2769 x 3, 24T2813 x 3, 24T2825, 24T2827, 32T96680, 36T1609, 36T1618 x 3, 36T1735 x 3, 36T1736 x 3, 36T1737 x 3, 36T1738 x 3, 36T1744 x 3, 36T1745 x 3, 36T1746 x 6, 36T1897 x 6, 36T1898 x 3, 36T1899 x 3, 36T1900 x 3, 36T1901 x 3, 36T1936 x 3, 36T1937 x 3, 36T1938 x 3, 36T1939Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy Classes
| Cycle Type | Size | Order | Representative |
| $ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ |
| $ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $( 3,12)( 6, 9)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1 $ | $18$ | $2$ | $( 1, 4)( 3,12)( 6, 9)( 7,10)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1 $ | $9$ | $2$ | $( 1, 4)( 3, 6)( 7,10)( 9,12)$ |
| $ 2, 2, 2, 2, 2, 2 $ | $9$ | $2$ | $( 1, 4)( 2,11)( 3,12)( 5, 8)( 6, 9)( 7,10)$ |
| $ 2, 2, 2, 2, 2, 2 $ | $3$ | $2$ | $( 1, 4)( 2,11)( 3, 6)( 5, 8)( 7,10)( 9,12)$ |
| $ 2, 2, 2, 2, 2, 2 $ | $9$ | $2$ | $( 1, 4)( 2,11)( 3, 9)( 5, 8)( 6,12)( 7,10)$ |
| $ 2, 2, 2, 2, 2, 2 $ | $6$ | $2$ | $( 1, 4)( 2, 8)( 3,12)( 5,11)( 6, 9)( 7,10)$ |
| $ 3, 3, 3, 1, 1, 1 $ | $64$ | $3$ | $( 4, 7,10)( 5,11, 8)( 6, 9,12)$ |
| $ 3, 3, 3, 1, 1, 1 $ | $64$ | $3$ | $( 4,10, 7)( 5, 8,11)( 6,12, 9)$ |
| $ 3, 3, 3, 3 $ | $32$ | $3$ | $( 1, 9, 5)( 2,10, 6)( 3,11, 7)( 4,12, 8)$ |
| $ 6, 6 $ | $96$ | $6$ | $( 1, 6, 2,10, 9, 5)( 3,11, 7,12, 8, 4)$ |
| $ 3, 3, 3, 3 $ | $32$ | $3$ | $( 1,12, 5)( 2, 4, 6)( 3, 8, 7)( 9,11,10)$ |
| $ 6, 6 $ | $96$ | $6$ | $( 1, 3, 8, 7,12, 5)( 2, 4, 9,11,10, 6)$ |
| $ 6, 6 $ | $96$ | $6$ | $( 1, 6, 2, 7, 3, 5)( 4, 9, 8,10,12,11)$ |
| $ 3, 3, 3, 3 $ | $32$ | $3$ | $( 1, 3, 5)( 2, 7, 6)( 4,12,11)( 8,10, 9)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1 $ | $12$ | $2$ | $( 1, 5)( 2,10)( 4, 8)( 7,11)$ |
| $ 2, 2, 2, 2, 2, 2 $ | $36$ | $2$ | $( 1, 5)( 2,10)( 3,12)( 4, 8)( 6, 9)( 7,11)$ |
| $ 4, 4, 1, 1, 1, 1 $ | $36$ | $4$ | $( 1, 5, 4, 8)( 2, 7,11,10)$ |
| $ 4, 4, 2, 2 $ | $36$ | $4$ | $( 1, 5, 4, 8)( 2, 7,11,10)( 3,12)( 6, 9)$ |
| $ 4, 4, 2, 2 $ | $36$ | $4$ | $( 1, 5, 4, 8)( 2, 7,11,10)( 3, 6)( 9,12)$ |
| $ 4, 4, 2, 2 $ | $36$ | $4$ | $( 1, 5, 4, 8)( 2, 7,11,10)( 3, 9)( 6,12)$ |
| $ 6, 3, 2, 1 $ | $192$ | $6$ | $( 1,11,10, 2, 4, 5)( 6, 9,12)( 7, 8)$ |
| $ 6, 3, 2, 1 $ | $192$ | $6$ | $( 1, 8,10, 2, 7, 5)( 4,11)( 6,12, 9)$ |
Group invariants
| Order: | $1152=2^{7} \cdot 3^{2}$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | Data not available |
| Character table: Data not available. |