Group action invariants
| Degree $n$ : | $21$ | |
| Transitive number $t$ : | $74$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,20,18)(2,21,16)(3,19,17)(4,14,6,13,5,15)(7,8,9)(10,11,12), (1,12,13,18)(2,11,14,17)(3,10,15,16)(4,6)(7,9)(19,21), (1,16,13,10)(2,17,14,11)(3,18,15,12)(7,19)(8,20)(9,21) | |
| $|\Aut(F/K)|$: | $1$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ x 3 4: $C_2^2$ 6: $S_3$ 12: $D_{6}$ 5040: $S_7$ 10080: $S_7\times C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $S_3$
Degree 7: $S_7$
Low degree siblings
42T845, 42T846, 42T847Siblings 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 $ | $1$ | $1$ | $()$ |
| $ 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1 $ | $3$ | $2$ | $( 2, 3)( 5, 6)( 8, 9)(11,12)(14,15)(17,18)(20,21)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ | $315$ | $2$ | $( 1,10)( 2,12)( 3,11)( 5, 6)( 7,16)( 8,18)( 9,17)(13,19)(14,21)(15,20)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1 $ | $105$ | $2$ | $( 1,10)( 2,11)( 3,12)( 7,16)( 8,17)( 9,18)(13,19)(14,20)(15,21)$ |
| $ 6, 6, 6, 3 $ | $210$ | $6$ | $( 1,11, 3,10, 2,12)( 4, 5, 6)( 7,17, 9,16, 8,18)(13,20,15,19,14,21)$ |
| $ 6, 6, 3, 3, 2, 1 $ | $840$ | $6$ | $( 1,17,13, 2,16,14)( 3,18,15)( 4, 5)( 7,20,10, 8,19,11)( 9,21,12)$ |
| $ 3, 3, 3, 3, 3, 3, 3 $ | $560$ | $3$ | $( 1,18,14)( 2,16,15)( 3,17,13)( 4, 6, 5)( 7,21,11)( 8,19,12)( 9,20,10)$ |
| $ 3, 3, 3, 3, 3, 3, 1, 1, 1 $ | $280$ | $3$ | $( 1,16,13)( 2,17,14)( 3,18,15)( 7,19,10)( 8,20,11)( 9,21,12)$ |
| $ 6, 6, 6, 2, 1 $ | $2520$ | $6$ | $( 1,20,16,11,13, 8)( 2,19,17,10,14, 7)( 3,21,18,12,15, 9)( 4, 5)$ |
| $ 6, 6, 6, 3 $ | $1680$ | $6$ | $( 1,21,17,10,15, 8)( 2,19,18,11,13, 9)( 3,20,16,12,14, 7)( 4, 6, 5)$ |
| $ 6, 6, 6, 1, 1, 1 $ | $840$ | $6$ | $( 1,19,16,10,13, 7)( 2,20,17,11,14, 8)( 3,21,18,12,15, 9)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ | $63$ | $2$ | $( 2, 3)( 5, 6)( 8, 9)(10,19)(11,21)(12,20)(14,15)(17,18)$ |
| $ 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $21$ | $2$ | $(10,19)(11,20)(12,21)$ |
| $ 6, 3, 3, 3, 3, 3 $ | $42$ | $6$ | $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,20,12,19,11,21)(13,14,15)(16,17,18)$ |
| $ 10, 5, 2, 2, 1, 1 $ | $1512$ | $10$ | $( 1,18, 4, 9,13, 3,16, 6, 7,15)( 2,17, 5, 8,14)(10,12)(19,21)$ |
| $ 15, 3, 3 $ | $1008$ | $15$ | $( 1,17, 6, 7,14, 3,16, 5, 9,13, 2,18, 4, 8,15)(10,11,12)(19,20,21)$ |
| $ 5, 5, 5, 1, 1, 1, 1, 1, 1 $ | $504$ | $5$ | $( 1,16, 4, 7,13)( 2,17, 5, 8,14)( 3,18, 6, 9,15)$ |
| $ 10, 5, 2, 2, 2 $ | $1512$ | $10$ | $( 1, 7,16,13, 4)( 2, 9,17,15, 5, 3, 8,18,14, 6)(10,19)(11,21)(12,20)$ |
| $ 5, 5, 5, 2, 2, 2 $ | $504$ | $10$ | $( 1, 7,16,13, 4)( 2, 8,17,14, 5)( 3, 9,18,15, 6)(10,19)(11,20)(12,21)$ |
| $ 15, 6 $ | $1008$ | $30$ | $( 1, 8,18,13, 5, 3, 7,17,15, 4, 2, 9,16,14, 6)(10,20,12,19,11,21)$ |
| $ 14, 7 $ | $2160$ | $14$ | $( 1,14,16,11, 4, 8,19, 2,13,17,10, 5, 7,20)( 3,15,18,12, 6, 9,21)$ |
| $ 21 $ | $1440$ | $21$ | $( 1,15,17,10, 6, 8,19, 3,14,16,12, 5, 7,21, 2,13,18,11, 4, 9,20)$ |
| $ 7, 7, 7 $ | $720$ | $7$ | $( 1,13,16,10, 4, 7,19)( 2,14,17,11, 5, 8,20)( 3,15,18,12, 6, 9,21)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $105$ | $2$ | $( 1,19)( 2,20)( 3,21)(13,16)(14,17)(15,18)$ |
| $ 6, 6, 3, 3, 3 $ | $210$ | $6$ | $( 1,20, 3,19, 2,21)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,17,15,16,14,18)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1 $ | $315$ | $2$ | $( 1,19)( 2,21)( 3,20)( 5, 6)( 8, 9)(11,12)(13,16)(14,18)(15,17)$ |
| $ 4, 4, 4, 2, 2, 2, 2, 1 $ | $1890$ | $4$ | $( 1,18,19,15)( 2,17,20,14)( 3,16,21,13)( 4, 6)( 7,12)( 8,11)( 9,10)$ |
| $ 12, 6, 3 $ | $1260$ | $12$ | $( 1,17,21,13, 2,18,19,14, 3,16,20,15)( 4, 5, 6)( 7,11, 9,10, 8,12)$ |
| $ 4, 4, 4, 2, 2, 2, 1, 1, 1 $ | $630$ | $4$ | $( 1,16,19,13)( 2,17,20,14)( 3,18,21,15)( 7,10)( 8,11)( 9,12)$ |
| $ 12, 3, 3, 3 $ | $420$ | $12$ | $( 1,15,20,16, 3,14,19,18, 2,13,21,17)( 4, 6, 5)( 7, 9, 8)(10,12,11)$ |
| $ 4, 4, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $210$ | $4$ | $( 1,13,19,16)( 2,14,20,17)( 3,15,21,18)$ |
| $ 4, 4, 4, 2, 2, 2, 1, 1, 1 $ | $630$ | $4$ | $( 1,14,19,17)( 2,13,20,16)( 3,15,21,18)( 4, 5)( 7, 8)(10,11)$ |
| $ 3, 3, 3, 3, 3, 3, 3 $ | $140$ | $3$ | $( 1, 2, 3)( 4,11, 9)( 5,12, 7)( 6,10, 8)(13,14,15)(16,17,18)(19,20,21)$ |
| $ 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $70$ | $3$ | $( 4,10, 7)( 5,11, 8)( 6,12, 9)$ |
| $ 6, 3, 2, 2, 2, 2, 1, 1, 1, 1 $ | $210$ | $6$ | $( 1, 3)( 4,12, 7, 6,10, 9)( 5,11, 8)(13,15)(16,18)(19,21)$ |
| $ 6, 6, 3, 3, 3 $ | $420$ | $6$ | $( 1,20, 3,19, 2,21)( 4, 8,12)( 5, 9,10)( 6, 7,11)(13,17,15,16,14,18)$ |
| $ 3, 3, 3, 2, 2, 2, 2, 2, 2 $ | $210$ | $6$ | $( 1,19)( 2,20)( 3,21)( 4, 7,10)( 5, 8,11)( 6, 9,12)(13,16)(14,17)(15,18)$ |
| $ 6, 3, 2, 2, 2, 2, 2, 2 $ | $630$ | $6$ | $( 1,21)( 2,20)( 3,19)( 4, 9,10, 6, 7,12)( 5, 8,11)(13,18)(14,17)(15,16)$ |
| $ 12, 3, 3, 3 $ | $840$ | $12$ | $( 1,15,17,19, 3,14,16,21, 2,13,18,20)( 4,12, 8)( 5,10, 9)( 6,11, 7)$ |
| $ 4, 4, 4, 3, 3, 3 $ | $420$ | $12$ | $( 1,13,16,19)( 2,14,17,20)( 3,15,18,21)( 4,10, 7)( 5,11, 8)( 6,12, 9)$ |
| $ 6, 4, 4, 4, 3 $ | $1260$ | $12$ | $( 1,14,16,20)( 2,13,17,19)( 3,15,18,21)( 4,11, 7, 5,10, 8)( 6,12, 9)$ |
| $ 6, 3, 2, 2, 2, 2, 2, 1, 1 $ | $1260$ | $6$ | $( 1, 6)( 2, 5)( 3, 4)( 7,15,16, 9,13,18)( 8,14,17)(10,12)(19,21)$ |
| $ 6, 3, 3, 3, 3, 3 $ | $840$ | $6$ | $( 1, 5, 3, 4, 2, 6)( 7,14,18)( 8,15,16)( 9,13,17)(10,11,12)(19,20,21)$ |
| $ 3, 3, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $420$ | $6$ | $( 1, 4)( 2, 5)( 3, 6)( 7,13,16)( 8,14,17)( 9,15,18)$ |
Group invariants
| Order: | $30240=2^{5} \cdot 3^{3} \cdot 5 \cdot 7$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | No | |
| GAP id: | Data not available |
| Character table: Data not available. |