Group action invariants
| Degree $n$ : | $12$ | |
| Transitive number $t$ : | $158$ | |
| CHM label : | $[2^{5}]F_{18}(6)$ | |
| Parity: | $1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,12)(2,3), (1,5,9)(4,8,12), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12) | |
| $|\Aut(F/K)|$: | $2$ |
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 12: $D_{6}$, $C_6\times C_2$ 18: $S_3\times C_3$ 36: $C_6\times S_3$ 288: $A_4\wr C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: None
Degree 4: None
Degree 6: $S_3\times C_3$
Low degree siblings
16T1028, 18T175 x 2, 24T1481 x 2, 24T1484 x 2, 24T1485, 24T1486, 32T34599 x 2, 36T720, 36T723, 36T743 x 2, 36T744 x 2, 36T948 x 2, 36T949, 36T950, 36T951 x 2Siblings 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 $ | $9$ | $2$ | $( 1,12)( 2, 3)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1 $ | $9$ | $2$ | $( 1,12)( 2, 3)( 6, 7)( 8, 9)$ |
| $ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $6$ | $2$ | $( 1,12)( 4, 5)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1 $ | $6$ | $2$ | $( 1,12)( 4, 5)( 6, 7)( 8, 9)$ |
| $ 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1,12)( 2, 3)( 4, 5)( 6, 7)( 8, 9)(10,11)$ |
| $ 3, 3, 1, 1, 1, 1, 1, 1 $ | $8$ | $3$ | $( 1, 5, 9)( 4, 8,12)$ |
| $ 6, 2, 1, 1, 1, 1 $ | $24$ | $6$ | $( 1, 5, 9,12, 4, 8)( 2, 3)$ |
| $ 3, 3, 2, 2, 1, 1 $ | $24$ | $6$ | $( 1, 5, 8)( 2, 3)( 4, 9,12)( 6, 7)$ |
| $ 6, 2, 2, 2 $ | $8$ | $6$ | $( 1, 5, 9,12, 4, 8)( 2, 3)( 6, 7)(10,11)$ |
| $ 3, 3, 1, 1, 1, 1, 1, 1 $ | $8$ | $3$ | $( 1, 9, 5)( 4,12, 8)$ |
| $ 6, 2, 1, 1, 1, 1 $ | $24$ | $6$ | $( 1, 9, 5,12, 8, 4)( 2, 3)$ |
| $ 3, 3, 2, 2, 1, 1 $ | $24$ | $6$ | $( 1, 8, 4)( 2, 3)( 5,12, 9)( 6, 7)$ |
| $ 6, 2, 2, 2 $ | $8$ | $6$ | $( 1, 9, 5,12, 8, 4)( 2, 3)( 6, 7)(10,11)$ |
| $ 3, 3, 3, 3 $ | $16$ | $3$ | $( 1, 5, 9)( 2, 6,10)( 3, 7,11)( 4, 8,12)$ |
| $ 6, 6 $ | $16$ | $6$ | $( 1, 5, 9,12, 4, 8)( 2, 6,10, 3, 7,11)$ |
| $ 3, 3, 3, 3 $ | $32$ | $3$ | $( 1, 9, 5)( 2, 6,10)( 3, 7,11)( 4,12, 8)$ |
| $ 6, 6 $ | $32$ | $6$ | $( 1, 9, 5,12, 8, 4)( 2, 6,10, 3, 7,11)$ |
| $ 3, 3, 3, 3 $ | $16$ | $3$ | $( 1, 9, 5)( 2,10, 6)( 3,11, 7)( 4,12, 8)$ |
| $ 6, 6 $ | $16$ | $6$ | $( 1, 9, 5,12, 8, 4)( 2,10, 6, 3,11, 7)$ |
| $ 2, 2, 2, 2, 2, 2 $ | $12$ | $2$ | $( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12)$ |
| $ 4, 4, 2, 2 $ | $36$ | $4$ | $( 1, 7,12, 6)( 2, 8, 3, 9)( 4,10)( 5,11)$ |
| $ 4, 4, 2, 2 $ | $36$ | $4$ | $( 1, 7,12, 6)( 2, 8)( 3, 9)( 4,10, 5,11)$ |
| $ 2, 2, 2, 2, 2, 2 $ | $12$ | $2$ | $( 1, 7)( 2, 8)( 3, 9)( 4,11)( 5,10)( 6,12)$ |
| $ 6, 6 $ | $48$ | $6$ | $( 1, 7, 5,11, 9, 3)( 2,12, 6, 4,10, 8)$ |
| $ 6, 6 $ | $48$ | $6$ | $( 1, 7, 5,11, 9, 2)( 3,12, 6, 4,10, 8)$ |
| $ 6, 6 $ | $48$ | $6$ | $( 1, 7, 9, 3, 5,11)( 2, 4,10,12, 6, 8)$ |
| $ 6, 6 $ | $48$ | $6$ | $( 1, 7, 9, 2, 4,10)( 3, 5,11,12, 6, 8)$ |
Group invariants
| Order: | $576=2^{6} \cdot 3^{2}$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | [576, 8656] |
| Character table: Data not available. |