Group action invariants
| Degree $n$ : | $20$ | |
| Transitive number $t$ : | $199$ | |
| Parity: | $1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,11,9,2,12,10)(3,20,6,8,14,15)(4,19,5,7,13,16)(17,18), (1,20,4,18)(2,19,3,17)(5,12)(6,11)(7,16,14,10)(8,15,13,9) | |
| $|\Aut(F/K)|$: | $2$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ x 3 4: $C_2^2$ 720: $S_6$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 4: None
Degree 5: None
Degree 10: $S_{6}$
Low degree siblings
12T219 x 4, 20T198 x 2, 20T199, 24T2959 x 2, 30T260 x 4, 30T261 x 4, 40T1177, 40T1178, 40T1189 x 2, 40T1190 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, 1, 1, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $15$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7,15)( 8,16)( 9,14)(10,13)(11,12)(17,20)(18,19)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $15$ | $2$ | $( 7,16)( 8,15)( 9,13)(10,14)(17,19)(18,20)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $45$ | $2$ | $( 5,12)( 6,11)( 7,10)( 8, 9)(13,15)(14,16)(17,19)(18,20)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $45$ | $2$ | $( 1, 2)( 3, 4)( 5,11)( 6,12)( 7, 9)( 8,10)(13,16)(14,15)(17,20)(18,19)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $15$ | $2$ | $( 5,18)( 6,17)( 7,10)( 8, 9)(11,19)(12,20)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $15$ | $2$ | $( 1, 2)( 3, 4)( 5,17)( 6,18)( 7, 9)( 8,10)(11,20)(12,19)(13,14)(15,16)$ |
| $ 4, 4, 4, 4, 2, 2 $ | $90$ | $4$ | $( 1, 2)( 3, 4)( 5,17,12,19)( 6,18,11,20)( 7,15,10,13)( 8,16, 9,14)$ |
| $ 4, 4, 4, 4, 1, 1, 1, 1 $ | $90$ | $4$ | $( 5,18,12,20)( 6,17,11,19)( 7,16,10,14)( 8,15, 9,13)$ |
| $ 6, 6, 3, 3, 1, 1 $ | $120$ | $6$ | $( 3, 6,10,16, 7,17)( 4, 5, 9,15, 8,18)(11,14,19)(12,13,20)$ |
| $ 6, 6, 6, 2 $ | $120$ | $6$ | $( 1, 2)( 3, 5,10,15, 7,18)( 4, 6, 9,16, 8,17)(11,13,19,12,14,20)$ |
| $ 3, 3, 3, 3, 3, 3, 1, 1 $ | $40$ | $3$ | $( 3, 6,11)( 4, 5,12)( 7,17,14)( 8,18,13)( 9,15,20)(10,16,19)$ |
| $ 6, 6, 6, 2 $ | $40$ | $6$ | $( 1, 2)( 3, 5,11, 4, 6,12)( 7,18,14, 8,17,13)( 9,16,20,10,15,19)$ |
| $ 6, 6, 6, 2 $ | $120$ | $6$ | $( 1, 2)( 3, 5,11, 4, 6,12)( 7,20,14,15,17, 9)( 8,19,13,16,18,10)$ |
| $ 6, 6, 3, 3, 1, 1 $ | $120$ | $6$ | $( 3, 6,11)( 4, 5,12)( 7,19,14,16,17,10)( 8,20,13,15,18, 9)$ |
| $ 3, 3, 3, 3, 3, 3, 1, 1 $ | $40$ | $3$ | $( 3, 7,10)( 4, 8, 9)( 5,18,15)( 6,17,16)(11,14,19)(12,13,20)$ |
| $ 6, 6, 6, 2 $ | $40$ | $6$ | $( 1, 2)( 3, 8,10, 4, 7, 9)( 5,17,15, 6,18,16)(11,13,19,12,14,20)$ |
| $ 4, 4, 4, 4, 2, 2 $ | $90$ | $4$ | $( 1, 4)( 2, 3)( 5,18,12,20)( 6,17,11,19)( 7,14,10,16)( 8,13, 9,15)$ |
| $ 4, 4, 4, 4, 2, 2 $ | $90$ | $4$ | $( 1, 3)( 2, 4)( 5,17,12,19)( 6,18,11,20)( 7,13,10,15)( 8,14, 9,16)$ |
| $ 10, 10 $ | $144$ | $10$ | $( 1, 3, 5, 7,18, 2, 4, 6, 8,17)( 9,14,20,11,15,10,13,19,12,16)$ |
| $ 5, 5, 5, 5 $ | $144$ | $5$ | $( 1, 4, 5, 8,18)( 2, 3, 6, 7,17)( 9,13,20,12,15)(10,14,19,11,16)$ |
Group invariants
| Order: | $1440=2^{5} \cdot 3^{2} \cdot 5$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | No | |
| GAP id: | [1440, 5842] |
| Character table: Data not available. |