Group action invariants
| Degree $n$ : | $21$ | |
| Transitive number $t$ : | $47$ | |
| Parity: | $1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,15,6,18)(2,17,5,16)(3,19,4,21)(7,20)(8,14,13,12,11,10,9), (1,9)(2,13,7,12)(3,10,6,8)(4,14,5,11)(15,17,19,21,16,18,20) | |
| $|\Aut(F/K)|$: | $1$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ 6: $S_3$ 24: $S_4$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $S_3$
Degree 7: None
Low degree siblings
28T348, 42T541, 42T542, 42T543, 42T547Siblings 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 $ | $1$ | $1$ | $()$ |
| $ 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $6$ | $7$ | $( 8,11,14,10,13, 9,12)$ |
| $ 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $6$ | $7$ | $( 8,10,12,14, 9,11,13)$ |
| $ 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $6$ | $7$ | $( 8,14,13,12,11,10, 9)$ |
| $ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ | $12$ | $7$ | $( 1, 2, 3, 4, 5, 6, 7)( 8,11,14,10,13, 9,12)$ |
| $ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ | $24$ | $7$ | $( 1, 2, 3, 4, 5, 6, 7)( 8,10,12,14, 9,11,13)$ |
| $ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ | $24$ | $7$ | $( 1, 2, 3, 4, 5, 6, 7)( 8,14,13,12,11,10, 9)$ |
| $ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ | $12$ | $7$ | $( 1, 4, 7, 3, 6, 2, 5)( 8,10,12,14, 9,11,13)$ |
| $ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ | $24$ | $7$ | $( 1, 4, 7, 3, 6, 2, 5)( 8,14,13,12,11,10, 9)$ |
| $ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ | $12$ | $7$ | $( 1, 3, 5, 7, 2, 4, 6)( 8,14,13,12,11,10, 9)$ |
| $ 7, 7, 7 $ | $8$ | $7$ | $( 1, 2, 3, 4, 5, 6, 7)( 8,11,14,10,13, 9,12)(15,17,19,21,16,18,20)$ |
| $ 7, 7, 7 $ | $24$ | $7$ | $( 1, 2, 3, 4, 5, 6, 7)( 8,10,12,14, 9,11,13)(15,17,19,21,16,18,20)$ |
| $ 7, 7, 7 $ | $24$ | $7$ | $( 1, 2, 3, 4, 5, 6, 7)( 8,14,13,12,11,10, 9)(15,17,19,21,16,18,20)$ |
| $ 7, 7, 7 $ | $24$ | $7$ | $( 1, 4, 7, 3, 6, 2, 5)( 8,10,12,14, 9,11,13)(15,17,19,21,16,18,20)$ |
| $ 7, 7, 7 $ | $24$ | $7$ | $( 1, 4, 7, 3, 6, 2, 5)( 8,14,13,12,11,10, 9)(15,17,19,21,16,18,20)$ |
| $ 7, 7, 7 $ | $24$ | $7$ | $( 1, 4, 7, 3, 6, 2, 5)( 8, 9,10,11,12,13,14)(15,17,19,21,16,18,20)$ |
| $ 7, 7, 7 $ | $24$ | $7$ | $( 1, 3, 5, 7, 2, 4, 6)( 8,14,13,12,11,10, 9)(15,17,19,21,16,18,20)$ |
| $ 7, 7, 7 $ | $8$ | $7$ | $( 1, 4, 7, 3, 6, 2, 5)( 8,10,12,14, 9,11,13)(15,21,20,19,18,17,16)$ |
| $ 7, 7, 7 $ | $24$ | $7$ | $( 1, 4, 7, 3, 6, 2, 5)( 8,14,13,12,11,10, 9)(15,21,20,19,18,17,16)$ |
| $ 7, 7, 7 $ | $24$ | $7$ | $( 1, 3, 5, 7, 2, 4, 6)( 8,14,13,12,11,10, 9)(15,21,20,19,18,17,16)$ |
| $ 7, 7, 7 $ | $8$ | $7$ | $( 1, 3, 5, 7, 2, 4, 6)( 8,14,13,12,11,10, 9)(15,19,16,20,17,21,18)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $147$ | $2$ | $( 2, 7)( 3, 6)( 4, 5)(16,21)(17,20)(18,19)$ |
| $ 7, 2, 2, 2, 2, 2, 2, 1, 1 $ | $294$ | $14$ | $( 2, 7)( 3, 6)( 4, 5)( 8,11,14,10,13, 9,12)(16,21)(17,20)(18,19)$ |
| $ 7, 2, 2, 2, 2, 2, 2, 1, 1 $ | $294$ | $14$ | $( 2, 7)( 3, 6)( 4, 5)( 8,10,12,14, 9,11,13)(16,21)(17,20)(18,19)$ |
| $ 7, 2, 2, 2, 2, 2, 2, 1, 1 $ | $294$ | $14$ | $( 2, 7)( 3, 6)( 4, 5)( 8,14,13,12,11,10, 9)(16,21)(17,20)(18,19)$ |
| $ 21 $ | $392$ | $21$ | $( 1,11,16, 7,14,18, 6,10,20, 5,13,15, 4, 9,17, 3,12,19, 2, 8,21)$ |
| $ 21 $ | $392$ | $21$ | $( 1,14,18, 6,13,15, 4,12,19, 2,11,16, 7,10,20, 5, 9,17, 3, 8,21)$ |
| $ 21 $ | $392$ | $21$ | $( 1,13,15, 4,11,16, 7, 9,17, 3,14,18, 6,12,19, 2,10,20, 5, 8,21)$ |
| $ 21 $ | $392$ | $21$ | $( 1,10,20, 5,12,19, 2,14,18, 6, 9,17, 3,11,16, 7,13,15, 4, 8,21)$ |
| $ 3, 3, 3, 3, 3, 3, 3 $ | $392$ | $3$ | $( 1, 8,21)( 2,12,19)( 3, 9,17)( 4,13,15)( 5,10,20)( 6,14,18)( 7,11,16)$ |
| $ 21 $ | $392$ | $21$ | $( 1, 9,17, 3,10,20, 5,11,16, 7,12,19, 2,13,15, 4,14,18, 6, 8,21)$ |
| $ 21 $ | $392$ | $21$ | $( 1,12,19, 2, 9,17, 3,13,15, 4,10,20, 5,14,18, 6,11,16, 7, 8,21)$ |
| $ 4, 4, 4, 2, 1, 1, 1, 1, 1, 1, 1 $ | $294$ | $4$ | $( 1,15, 6,18)( 2,17, 5,16)( 3,19, 4,21)( 7,20)$ |
| $ 7, 4, 4, 4, 2 $ | $294$ | $28$ | $( 1,15, 6,18)( 2,17, 5,16)( 3,19, 4,21)( 7,20)( 8,11,14,10,13, 9,12)$ |
| $ 7, 4, 4, 4, 2 $ | $294$ | $28$ | $( 1,15, 6,18)( 2,17, 5,16)( 3,19, 4,21)( 7,20)( 8,10,12,14, 9,11,13)$ |
| $ 7, 4, 4, 4, 2 $ | $294$ | $28$ | $( 1,15, 6,18)( 2,17, 5,16)( 3,19, 4,21)( 7,20)( 8,14,13,12,11,10, 9)$ |
| $ 7, 4, 4, 4, 2 $ | $294$ | $28$ | $( 1,15, 6,18)( 2,17, 5,16)( 3,19, 4,21)( 7,20)( 8,12, 9,13,10,14,11)$ |
| $ 7, 4, 4, 4, 2 $ | $294$ | $28$ | $( 1,15, 6,18)( 2,17, 5,16)( 3,19, 4,21)( 7,20)( 8,13,11, 9,14,12,10)$ |
| $ 7, 4, 4, 4, 2 $ | $294$ | $28$ | $( 1,15, 6,18)( 2,17, 5,16)( 3,19, 4,21)( 7,20)( 8, 9,10,11,12,13,14)$ |
| $ 14, 2, 2, 2, 1 $ | $588$ | $14$ | $( 1,15, 6,19, 4,16, 2,20, 7,17, 5,21, 3,18)( 9,14)(10,13)(11,12)$ |
| $ 14, 2, 2, 2, 1 $ | $588$ | $14$ | $( 1,15, 7,17, 6,19, 5,21, 4,16, 3,18, 2,20)( 9,14)(10,13)(11,12)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ | $294$ | $2$ | $( 1,15)( 2,20)( 3,18)( 4,16)( 5,21)( 6,19)( 7,17)( 9,14)(10,13)(11,12)$ |
| $ 14, 2, 2, 2, 1 $ | $588$ | $14$ | $( 1,15, 5,21, 2,20, 6,19, 3,18, 7,17, 4,16)( 9,14)(10,13)(11,12)$ |
Group invariants
| Order: | $8232=2^{3} \cdot 3 \cdot 7^{3}$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | Data not available |
| Character table: Data not available. |