Group action invariants
| Degree $n$ : | $12$ | |
| Transitive number $t$ : | $205$ | |
| CHM label : | $[E(4)^{3}:3:2]3$ | |
| 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), (1,10)(2,5)(6,9), (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: $S_3$, $C_6$ 18: $S_3\times C_3$ 24: $S_4$ 72: 12T45 288: $A_4\wr C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 3: $C_3$
Degree 4: None
Degree 6: None
Low degree siblings
18T269, 18T270, 24T2765, 24T2767, 24T2812, 24T2823, 24T2829, 24T2831, 32T96681, 36T1610, 36T1615, 36T1617, 36T1724, 36T1725, 36T1755, 36T1756, 36T1757, 36T1896 x 2, 36T1902, 36T1903, 36T1904, 36T1905, 36T1940, 36T1941, 36T1942 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 $ | $9$ | $2$ | $( 3,12)( 6, 9)$ |
| $ 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 $ | $18$ | $2$ | $( 1, 4)( 2,11)( 3,12)( 5, 8)( 6, 9)( 7,10)$ |
| $ 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 $ | $6$ | $2$ | $( 1, 4)( 2, 8)( 3,12)( 5,11)( 6, 9)( 7,10)$ |
| $ 3, 3, 3, 1, 1, 1 $ | $128$ | $3$ | $( 4, 7,10)( 5,11, 8)( 6, 9,12)$ |
| $ 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $24$ | $2$ | $( 5, 8)( 7,10)( 9,12)$ |
| $ 4, 2, 2, 1, 1, 1, 1 $ | $72$ | $4$ | $( 3,12, 6, 9)( 5, 8)( 7,10)$ |
| $ 4, 4, 2, 1, 1 $ | $72$ | $4$ | $( 2, 8,11, 5)( 3,12, 6, 9)( 7,10)$ |
| $ 4, 4, 4 $ | $24$ | $4$ | $( 1, 7, 4,10)( 2, 8,11, 5)( 3,12, 6, 9)$ |
| $ 3, 3, 3, 3 $ | $16$ | $3$ | $( 1, 5, 9)( 2, 6,10)( 3, 7,11)( 4, 8,12)$ |
| $ 6, 6 $ | $48$ | $6$ | $( 1, 5, 6,10, 2, 9)( 3, 7,11,12, 4, 8)$ |
| $ 6, 6 $ | $96$ | $6$ | $( 1,11, 3,10, 2, 9)( 4, 5,12, 7, 8, 6)$ |
| $ 3, 3, 3, 3 $ | $32$ | $3$ | $( 1,11, 9)( 2, 3,10)( 4, 5, 6)( 7, 8,12)$ |
| $ 6, 3, 3 $ | $96$ | $6$ | $( 1, 8, 9)( 2, 6, 7,11, 3,10)( 4, 5,12)$ |
| $ 12 $ | $96$ | $12$ | $( 1, 8, 6, 7,11,12, 4, 5, 3,10, 2, 9)$ |
| $ 3, 3, 3, 3 $ | $16$ | $3$ | $( 1, 9, 5)( 2,10, 6)( 3,11, 7)( 4,12, 8)$ |
| $ 6, 6 $ | $48$ | $6$ | $( 1, 6, 2,10, 9, 5)( 3,11, 7,12, 8, 4)$ |
| $ 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, 3, 3 $ | $96$ | $6$ | $( 1,12, 5)( 2, 7, 3,11,10, 6)( 4, 9, 8)$ |
| $ 12 $ | $96$ | $12$ | $( 1, 3,11,10, 9, 8, 4, 6, 2, 7,12, 5)$ |
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. |