Group action invariants
| Degree $n$ : | $20$ | |
| Transitive number $t$ : | $168$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,15,7,12,4,18,9,14,5,19)(2,16,8,11,3,17,10,13,6,20), (1,19,3,11)(2,20,4,12)(5,14,10,17)(6,13,9,18)(7,15,8,16), (1,11,10,14,7,16,6,18,4,20,2,12,9,13,8,15,5,17,3,19) | |
| $|\Aut(F/K)|$: | $2$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ x 7 4: $C_2^2$ x 7 8: $D_{4}$ x 6, $C_2^3$ 16: $D_4\times C_2$ x 3 32: $C_2^2 \wr C_2$ 200: $D_5^2 : C_2$ 400: 20T92 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 4: $D_{4}$
Degree 5: None
Degree 10: $D_5^2 : C_2$
Low degree siblings
20T168 x 7, 40T615 x 4, 40T622 x 2, 40T628 x 4, 40T651 x 2, 40T660 x 4, 40T705 x 8, 40T765 x 2, 40T797 x 2, 40T802 x 4, 40T803 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 $ | $1$ | $1$ | $()$ |
| $ 5, 5, 5, 5 $ | $4$ | $5$ | $( 1, 7, 4, 9, 5)( 2, 8, 3,10, 6)(11,17,13,20,16)(12,18,14,19,15)$ |
| $ 5, 5, 5, 5 $ | $4$ | $5$ | $( 1, 4, 5, 7, 9)( 2, 3, 6, 8,10)(11,13,16,17,20)(12,14,15,18,19)$ |
| $ 5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $4$ | $5$ | $(11,13,16,17,20)(12,14,15,18,19)$ |
| $ 5, 5, 5, 5 $ | $8$ | $5$ | $( 1, 7, 4, 9, 5)( 2, 8, 3,10, 6)(11,20,17,16,13)(12,19,18,15,14)$ |
| $ 5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $4$ | $5$ | $(11,16,20,13,17)(12,15,19,14,18)$ |
| $ 10, 10 $ | $40$ | $10$ | $( 1,15, 7,12, 4,18, 9,14, 5,19)( 2,16, 8,11, 3,17,10,13, 6,20)$ |
| $ 10, 10 $ | $40$ | $10$ | $( 1,12, 9,19, 7,18, 5,15, 4,14)( 2,11,10,20, 8,17, 6,16, 3,13)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $20$ | $2$ | $( 1,18)( 2,17)( 3,20)( 4,19)( 5,12)( 6,11)( 7,14)( 8,13)( 9,15)(10,16)$ |
| $ 10, 10 $ | $4$ | $10$ | $( 1, 3, 5, 8, 9, 2, 4, 6, 7,10)(11,19,17,15,13,12,20,18,16,14)$ |
| $ 10, 10 $ | $8$ | $10$ | $( 1,10, 7, 6, 4, 2, 9, 8, 5, 3)(11,15,20,14,17,12,16,19,13,18)$ |
| $ 10, 2, 2, 2, 2, 2 $ | $4$ | $10$ | $( 1, 6, 9, 3, 7, 2, 5,10, 4, 8)(11,12)(13,14)(15,16)(17,18)(19,20)$ |
| $ 10, 2, 2, 2, 2, 2 $ | $4$ | $10$ | $( 1, 3, 5, 8, 9, 2, 4, 6, 7,10)(11,12)(13,14)(15,16)(17,18)(19,20)$ |
| $ 10, 10 $ | $4$ | $10$ | $( 1, 8, 4,10, 5, 2, 7, 3, 9, 6)(11,15,20,14,17,12,16,19,13,18)$ |
| $ 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, 1, 1, 1, 1 $ | $25$ | $2$ | $( 3,10)( 4, 9)( 5, 7)( 6, 8)(13,20)(14,19)(15,18)(16,17)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $25$ | $2$ | $( 1, 3)( 2, 4)( 5,10)( 6, 9)( 7, 8)(11,19)(12,20)(13,18)(14,17)(15,16)$ |
| $ 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $10$ | $2$ | $(11,15)(12,16)(13,14)(17,19)(18,20)$ |
| $ 5, 5, 2, 2, 2, 2, 2 $ | $20$ | $10$ | $( 1, 7, 4, 9, 5)( 2, 8, 3,10, 6)(11,12)(13,19)(14,20)(15,17)(16,18)$ |
| $ 5, 5, 2, 2, 2, 2, 2 $ | $20$ | $10$ | $( 1, 4, 5, 7, 9)( 2, 3, 6, 8,10)(11,18)(12,17)(13,15)(14,16)(19,20)$ |
| $ 4, 4, 4, 4, 4 $ | $100$ | $4$ | $( 1,15, 3,17)( 2,16, 4,18)( 5,19,10,13)( 6,20, 9,14)( 7,12, 8,11)$ |
| $ 10, 2, 2, 2, 2, 1, 1 $ | $20$ | $10$ | $( 1, 3, 5, 8, 9, 2, 4, 6, 7,10)(11,13)(12,14)(15,19)(16,20)$ |
| $ 10, 2, 2, 2, 2, 1, 1 $ | $20$ | $10$ | $( 1, 6, 9, 3, 7, 2, 5,10, 4, 8)(11,16)(12,15)(17,20)(18,19)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $10$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(13,20)(14,19)(15,18)(16,17)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $10$ | $2$ | $(13,20)(14,19)(15,18)(16,17)$ |
| $ 5, 5, 2, 2, 2, 2, 1, 1 $ | $20$ | $10$ | $( 1, 7, 4, 9, 5)( 2, 8, 3,10, 6)(11,17)(12,18)(13,16)(14,15)$ |
| $ 5, 5, 2, 2, 2, 2, 1, 1 $ | $20$ | $10$ | $( 1, 4, 5, 7, 9)( 2, 3, 6, 8,10)(11,13)(12,14)(15,19)(16,20)$ |
| $ 4, 4, 4, 4, 2, 2 $ | $100$ | $4$ | $( 1,15, 9,14)( 2,16,10,13)( 3,17, 8,11)( 4,18, 7,12)( 5,19)( 6,20)$ |
| $ 10, 2, 2, 2, 2, 2 $ | $20$ | $10$ | $( 1, 3, 5, 8, 9, 2, 4, 6, 7,10)(11,19)(12,20)(13,18)(14,17)(15,16)$ |
| $ 10, 2, 2, 2, 2, 2 $ | $20$ | $10$ | $( 1, 6, 9, 3, 7, 2, 5,10, 4, 8)(11,12)(13,19)(14,20)(15,17)(16,18)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $10$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,18)(12,17)(13,15)(14,16)(19,20)$ |
| $ 10, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $4$ | $10$ | $(11,15,20,14,17,12,16,19,13,18)$ |
| $ 5, 5, 2, 2, 2, 2, 2 $ | $4$ | $10$ | $( 1, 7, 4, 9, 5)( 2, 8, 3,10, 6)(11,12)(13,14)(15,16)(17,18)(19,20)$ |
| $ 10, 5, 5 $ | $8$ | $10$ | $( 1, 4, 5, 7, 9)( 2, 3, 6, 8,10)(11,18,13,19,16,12,17,14,20,15)$ |
| $ 10, 5, 5 $ | $8$ | $10$ | $( 1, 5, 9, 4, 7)( 2, 6,10, 3, 8)(11,19,17,15,13,12,20,18,16,14)$ |
| $ 10, 5, 5 $ | $8$ | $10$ | $( 1, 9, 7, 5, 4)( 2,10, 8, 6, 3)(11,14,16,18,20,12,13,15,17,19)$ |
| $ 10, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $4$ | $10$ | $(11,19,17,15,13,12,20,18,16,14)$ |
| $ 10, 5, 5 $ | $8$ | $10$ | $( 1, 7, 4, 9, 5)( 2, 8, 3,10, 6)(11,15,20,14,17,12,16,19,13,18)$ |
| $ 5, 5, 2, 2, 2, 2, 2 $ | $4$ | $10$ | $( 1, 4, 5, 7, 9)( 2, 3, 6, 8,10)(11,12)(13,14)(15,16)(17,18)(19,20)$ |
| $ 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $2$ | $2$ | $(11,12)(13,14)(15,16)(17,18)(19,20)$ |
| $ 4, 4, 4, 4, 4 $ | $20$ | $4$ | $( 1,15, 2,16)( 3,17, 4,18)( 5,19, 6,20)( 7,12, 8,11)( 9,14,10,13)$ |
| $ 20 $ | $40$ | $20$ | $( 1,12, 3,13, 5,15, 8,17, 9,19, 2,11, 4,14, 6,16, 7,18,10,20)$ |
| $ 20 $ | $40$ | $20$ | $( 1,18, 6,11, 9,15, 3,20, 7,14, 2,17, 5,12,10,16, 4,19, 8,13)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $50$ | $2$ | $( 3,10)( 4, 9)( 5, 7)( 6, 8)(11,18)(12,17)(13,15)(14,16)(19,20)$ |
Group invariants
| Order: | $800=2^{5} \cdot 5^{2}$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | [800, 1058] |
| Character table: Data not available. |