Group action invariants
| Degree $n$ : | $21$ | |
| Transitive number $t$ : | $56$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,4,17)(2,5,18)(3,6,16)(7,14,19,12,9,13,21,11,8,15,20,10), (4,11,14,18,20,9)(5,12,15,16,21,7)(6,10,13,17,19,8) | |
| $|\Aut(F/K)|$: | $3$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ 3: $C_3$ 6: $C_6$ 5040: $S_7$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $C_3$
Degree 7: $S_7$
Low degree siblings
42T666Siblings 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 $ | $1$ | $3$ | $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)$ |
| $ 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,12,11)(13,15,14)(16,18,17)(19,21,20)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $105$ | $2$ | $( 4,20)( 5,21)( 6,19)(13,17)(14,18)(15,16)$ |
| $ 6, 6, 3, 3, 3 $ | $105$ | $6$ | $( 1, 2, 3)( 4,21, 6,20, 5,19)( 7, 8, 9)(10,11,12)(13,18,15,17,14,16)$ |
| $ 6, 6, 3, 3, 3 $ | $105$ | $6$ | $( 1, 3, 2)( 4,19, 5,20, 6,21)( 7, 9, 8)(10,12,11)(13,16,14,17,15,18)$ |
| $ 12, 3, 3, 3 $ | $210$ | $12$ | $( 1, 2, 3)( 4,15,19,18, 5,13,20,16, 6,14,21,17)( 7, 8, 9)(10,11,12)$ |
| $ 12, 3, 3, 3 $ | $210$ | $12$ | $( 1, 3, 2)( 4,13,21,18, 6,15,20,17, 5,14,19,16)( 7, 9, 8)(10,12,11)$ |
| $ 4, 4, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $210$ | $4$ | $( 4,14,20,18)( 5,15,21,16)( 6,13,19,17)$ |
| $ 6, 6, 6, 3 $ | $105$ | $6$ | $( 1,10, 3,12, 2,11)( 4,21, 6,20, 5,19)( 7, 8, 9)(13,18,15,17,14,16)$ |
| $ 6, 6, 6, 3 $ | $105$ | $6$ | $( 1,11, 2,12, 3,10)( 4,19, 5,20, 6,21)( 7, 9, 8)(13,16,14,17,15,18)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1 $ | $105$ | $2$ | $( 1,12)( 2,10)( 3,11)( 4,20)( 5,21)( 6,19)(13,17)(14,18)(15,16)$ |
| $ 3, 3, 3, 3, 3, 3, 3 $ | $280$ | $3$ | $( 1, 6,18)( 2, 4,16)( 3, 5,17)( 7, 8, 9)(10,20,15)(11,21,13)(12,19,14)$ |
| $ 3, 3, 3, 3, 3, 3, 3 $ | $280$ | $3$ | $( 1, 4,17)( 2, 5,18)( 3, 6,16)( 7, 9, 8)(10,21,14)(11,19,15)(12,20,13)$ |
| $ 3, 3, 3, 3, 3, 3, 1, 1, 1 $ | $280$ | $3$ | $( 1, 5,16)( 2, 6,17)( 3, 4,18)(10,19,13)(11,20,14)(12,21,15)$ |
| $ 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $70$ | $3$ | $(10,19,13)(11,20,14)(12,21,15)$ |
| $ 3, 3, 3, 3, 3, 3, 3 $ | $70$ | $3$ | $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,20,15)(11,21,13)(12,19,14)(16,17,18)$ |
| $ 3, 3, 3, 3, 3, 3, 3 $ | $70$ | $3$ | $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,21,14)(11,19,15)(12,20,13)(16,18,17)$ |
| $ 6, 6, 6, 3 $ | $840$ | $6$ | $( 1,20,17,12, 4,13)( 2,21,18,10, 5,14)( 3,19,16,11, 6,15)( 7, 9, 8)$ |
| $ 6, 6, 6, 1, 1, 1 $ | $840$ | $6$ | $( 1,21,16,12, 5,15)( 2,19,17,10, 6,13)( 3,20,18,11, 4,14)$ |
| $ 6, 6, 6, 3 $ | $840$ | $6$ | $( 1,19,18,12, 6,14)( 2,20,16,10, 4,15)( 3,21,17,11, 5,13)( 7, 8, 9)$ |
| $ 6, 6, 3, 3, 3 $ | $210$ | $6$ | $( 1,19,18)( 2,20,16)( 3,21,17)( 4, 7, 6, 9, 5, 8)(10,14,12,13,11,15)$ |
| $ 6, 6, 3, 3, 3 $ | $210$ | $6$ | $( 1,20,17)( 2,21,18)( 3,19,16)( 4, 8, 5, 9, 6, 7)(10,15,11,13,12,14)$ |
| $ 3, 3, 3, 2, 2, 2, 2, 2, 2 $ | $210$ | $6$ | $( 1,21,16)( 2,19,17)( 3,20,18)( 4, 9)( 5, 7)( 6, 8)(10,13)(11,14)(12,15)$ |
| $ 4, 4, 4, 3, 3, 3 $ | $420$ | $12$ | $( 1,16,21)( 2,17,19)( 3,18,20)( 4,11, 9,14)( 5,12, 7,15)( 6,10, 8,13)$ |
| $ 12, 3, 3, 3 $ | $420$ | $12$ | $( 1,17,20)( 2,18,21)( 3,16,19)( 4,12, 8,14, 5,10, 9,15, 6,11, 7,13)$ |
| $ 12, 3, 3, 3 $ | $420$ | $12$ | $( 1,18,19)( 2,16,20)( 3,17,21)( 4,10, 7,14, 6,12, 9,13, 5,11, 8,15)$ |
| $ 4, 4, 4, 2, 2, 2, 1, 1, 1 $ | $630$ | $4$ | $( 4,14,20,11)( 5,15,21,12)( 6,13,19,10)( 7,16)( 8,17)( 9,18)$ |
| $ 12, 6, 3 $ | $630$ | $12$ | $( 1, 2, 3)( 4,15,19,11, 5,13,20,12, 6,14,21,10)( 7,17, 9,16, 8,18)$ |
| $ 12, 6, 3 $ | $630$ | $12$ | $( 1, 3, 2)( 4,13,21,11, 6,15,20,10, 5,14,19,12)( 7,18, 8,16, 9,17)$ |
| $ 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $21$ | $2$ | $( 7,15)( 8,13)( 9,14)$ |
| $ 6, 3, 3, 3, 3, 3 $ | $21$ | $6$ | $( 1, 2, 3)( 4, 5, 6)( 7,13, 9,15, 8,14)(10,11,12)(16,17,18)(19,20,21)$ |
| $ 6, 3, 3, 3, 3, 3 $ | $21$ | $6$ | $( 1, 3, 2)( 4, 6, 5)( 7,14, 8,15, 9,13)(10,12,11)(16,18,17)(19,21,20)$ |
| $ 5, 5, 5, 1, 1, 1, 1, 1, 1 $ | $504$ | $5$ | $( 1,12,16,21, 5)( 2,10,17,19, 6)( 3,11,18,20, 4)$ |
| $ 15, 3, 3 $ | $504$ | $15$ | $( 1,10,18,21, 6, 3,12,17,20, 5, 2,11,16,19, 4)( 7, 8, 9)(13,14,15)$ |
| $ 15, 3, 3 $ | $504$ | $15$ | $( 1,11,17,21, 4, 2,12,18,19, 5, 3,10,16,20, 6)( 7, 9, 8)(13,15,14)$ |
| $ 15, 6 $ | $504$ | $30$ | $( 1,20,10, 5,18, 2,21,11, 6,16, 3,19,12, 4,17)( 7,14, 8,15, 9,13)$ |
| $ 5, 5, 5, 2, 2, 2 $ | $504$ | $10$ | $( 1,21,12, 5,16)( 2,19,10, 6,17)( 3,20,11, 4,18)( 7,15)( 8,13)( 9,14)$ |
| $ 15, 6 $ | $504$ | $30$ | $( 1,19,11, 5,17, 3,21,10, 4,16, 2,20,12, 6,18)( 7,13, 9,15, 8,14)$ |
| $ 6, 3, 3, 3, 3, 3 $ | $420$ | $6$ | $( 1, 3, 2)( 4,19,16)( 5,20,17)( 6,21,18)( 7,14, 8,15, 9,13)(10,12,11)$ |
| $ 3, 3, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $420$ | $6$ | $( 4,20,18)( 5,21,16)( 6,19,17)( 7,15)( 8,13)( 9,14)$ |
| $ 6, 3, 3, 3, 3, 3 $ | $420$ | $6$ | $( 1, 2, 3)( 4,21,17)( 5,19,18)( 6,20,16)( 7,13, 9,15, 8,14)(10,11,12)$ |
| $ 21 $ | $720$ | $21$ | $( 1,10, 9, 5,17,20,15, 2,11, 7, 6,18,21,13, 3,12, 8, 4,16,19,14)$ |
| $ 21 $ | $720$ | $21$ | $( 1,11, 8, 5,18,19,15, 3,10, 7, 4,17,21,14, 2,12, 9, 6,16,20,13)$ |
| $ 7, 7, 7 $ | $720$ | $7$ | $( 1,12, 7, 5,16,21,15)( 2,10, 8, 6,17,19,13)( 3,11, 9, 4,18,20,14)$ |
Group invariants
| Order: | $15120=2^{4} \cdot 3^{3} \cdot 5 \cdot 7$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | No | |
| GAP id: | Data not available |
| Character table: Data not available. |