Group action invariants
| Degree $n$ : | $12$ | |
| Transitive number $t$ : | $240$ | |
| CHM label : | $[2^{6}]F_{18}:2=2wrF_{18}(6):2$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,12), (1,5,9)(4,8,12), (2,10)(3,11)(4,8)(5,9), (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 7 4: $C_2^2$ x 7 6: $S_3$ x 2 8: $D_{4}$ x 2, $C_2^3$ 12: $D_{6}$ x 6 16: $D_4\times C_2$ 24: $S_3 \times C_2^2$ x 2 36: $S_3^2$ 48: 12T28 x 2 72: 12T37 144: 12T81 576: $(A_4\wr C_2):C_2$ 1152: 12T195 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: None
Degree 4: None
Degree 6: $S_3^2$
Low degree siblings
12T240 x 3, 16T1498 x 4, 24T5064 x 2, 24T5132 x 2, 24T5133 x 4, 24T5134 x 4, 24T5135 x 4, 24T5136 x 4, 24T5137 x 2, 24T5138 x 2, 32T205440 x 2, 32T205441 x 2, 32T205442 x 2, 36T3098 x 2, 36T3100 x 4, 36T3163 x 2, 36T3436 x 4Siblings 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $6$ | $2$ | $( 1,12)$ |
| $ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $( 1,12)( 6, 7)$ |
| $ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $6$ | $2$ | $( 1,12)( 4, 5)$ |
| $ 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $18$ | $2$ | $( 1,12)( 4, 5)( 6, 7)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1 $ | $9$ | $2$ | $( 1,12)( 4, 5)( 6, 7)(10,11)$ |
| $ 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $2$ | $2$ | $( 1,12)( 4, 5)( 8, 9)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1 $ | $6$ | $2$ | $( 1,12)( 4, 5)( 6, 7)( 8, 9)$ |
| $ 2, 2, 2, 2, 2, 1, 1 $ | $6$ | $2$ | $( 1,12)( 4, 5)( 6, 7)( 8, 9)(10,11)$ |
| $ 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 $ | $16$ | $3$ | $( 1, 5, 9)( 4, 8,12)$ |
| $ 6, 1, 1, 1, 1, 1, 1 $ | $16$ | $6$ | $( 1, 5, 9,12, 4, 8)$ |
| $ 3, 3, 2, 1, 1, 1, 1 $ | $48$ | $6$ | $( 1, 5, 9)( 4, 8,12)( 6, 7)$ |
| $ 6, 2, 1, 1, 1, 1 $ | $48$ | $6$ | $( 1, 5, 9,12, 4, 8)( 6, 7)$ |
| $ 3, 3, 2, 2, 1, 1 $ | $48$ | $6$ | $( 1, 5, 9)( 4, 8,12)( 6, 7)(10,11)$ |
| $ 6, 2, 2, 1, 1 $ | $48$ | $6$ | $( 1, 5, 9,12, 4, 8)( 6, 7)(10,11)$ |
| $ 3, 3, 2, 2, 2 $ | $16$ | $6$ | $( 1, 5, 9)( 2, 3)( 4, 8,12)( 6, 7)(10,11)$ |
| $ 6, 2, 2, 2 $ | $16$ | $6$ | $( 1, 5, 9,12, 4, 8)( 2, 3)( 6, 7)(10,11)$ |
| $ 3, 3, 3, 3 $ | $32$ | $3$ | $( 1, 5, 9)( 2, 6,10)( 3, 7,11)( 4, 8,12)$ |
| $ 6, 3, 3 $ | $64$ | $6$ | $( 1, 5, 9,12, 4, 8)( 2, 6,10)( 3, 7,11)$ |
| $ 6, 6 $ | $32$ | $6$ | $( 1, 5, 9,12, 4, 8)( 2, 7,11, 3, 6,10)$ |
| $ 3, 3, 3, 3 $ | $32$ | $3$ | $( 1, 9, 5)( 2, 6,10)( 3, 7,11)( 4,12, 8)$ |
| $ 6, 3, 3 $ | $64$ | $6$ | $( 1, 9, 5,12, 8, 4)( 2, 6,10)( 3, 7,11)$ |
| $ 6, 6 $ | $32$ | $6$ | $( 1, 9, 5,12, 8, 4)( 2, 7,11, 3, 6,10)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1 $ | $36$ | $2$ | $( 2,10)( 3,11)( 4, 8)( 5, 9)$ |
| $ 2, 2, 2, 2, 2, 1, 1 $ | $72$ | $2$ | $( 1,12)( 2,10)( 3,11)( 4, 8)( 5, 9)$ |
| $ 2, 2, 2, 2, 2, 2 $ | $36$ | $2$ | $( 1,12)( 2,10)( 3,11)( 4, 8)( 5, 9)( 6, 7)$ |
| $ 4, 2, 2, 1, 1, 1, 1 $ | $72$ | $4$ | $( 2,10)( 3,11)( 4, 8, 5, 9)$ |
| $ 4, 2, 2, 2, 1, 1 $ | $72$ | $4$ | $( 1,12)( 2,10)( 3,11)( 4, 8, 5, 9)$ |
| $ 4, 2, 2, 2, 1, 1 $ | $72$ | $4$ | $( 2,10)( 3,11)( 4, 8, 5, 9)( 6, 7)$ |
| $ 4, 2, 2, 2, 2 $ | $72$ | $4$ | $( 1,12)( 2,10)( 3,11)( 4, 8, 5, 9)( 6, 7)$ |
| $ 4, 4, 1, 1, 1, 1 $ | $36$ | $4$ | $( 2,11, 3,10)( 4, 8, 5, 9)$ |
| $ 4, 4, 2, 1, 1 $ | $72$ | $4$ | $( 1,12)( 2,11, 3,10)( 4, 8, 5, 9)$ |
| $ 4, 4, 2, 2 $ | $36$ | $4$ | $( 1,12)( 2,11, 3,10)( 4, 8, 5, 9)( 6, 7)$ |
| $ 2, 2, 2, 2, 2, 2 $ | $24$ | $2$ | $( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12)$ |
| $ 4, 2, 2, 2, 2 $ | $72$ | $4$ | $( 1, 7,12, 6)( 2, 8)( 3, 9)( 4,10)( 5,11)$ |
| $ 4, 4, 2, 2 $ | $72$ | $4$ | $( 1, 7,12, 6)( 2, 8)( 3, 9)( 4,10, 5,11)$ |
| $ 4, 4, 4 $ | $24$ | $4$ | $( 1, 7,12, 6)( 2, 9, 3, 8)( 4,10, 5,11)$ |
| $ 6, 6 $ | $192$ | $6$ | $( 1, 7, 5,11, 9, 3)( 2,12, 6, 4,10, 8)$ |
| $ 12 $ | $192$ | $12$ | $( 1, 7, 5,11, 9, 3,12, 6, 4,10, 8, 2)$ |
| $ 2, 2, 2, 2, 2, 2 $ | $24$ | $2$ | $( 1, 7)( 2, 4)( 3, 5)( 6,12)( 8,10)( 9,11)$ |
| $ 4, 2, 2, 2, 2 $ | $72$ | $4$ | $( 1, 7,12, 6)( 2, 4)( 3, 5)( 8,10)( 9,11)$ |
| $ 4, 4, 2, 2 $ | $72$ | $4$ | $( 1, 7,12, 6)( 2, 5, 3, 4)( 8,10)( 9,11)$ |
| $ 4, 4, 4 $ | $24$ | $4$ | $( 1, 7,12, 6)( 2, 5, 3, 4)( 8,11, 9,10)$ |
| $ 6, 6 $ | $192$ | $6$ | $( 1, 7, 5, 3, 9,11)( 2, 8,10,12, 6, 4)$ |
| $ 12 $ | $192$ | $12$ | $( 1, 7, 5, 3, 9,11,12, 6, 4, 2, 8,10)$ |
Group invariants
| Order: | $2304=2^{8} \cdot 3^{2}$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | Data not available |
| Character table: Data not available. |