Group action invariants
| Degree $n$ : | $21$ | |
| Transitive number $t$ : | $70$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,5,3,4,7,2)(8,14,10)(9,11,12)(15,16,19,21,20,17), (1,10,20,6,11,15)(2,13,19,5,8,16)(3,9,18,4,12,17)(7,14,21) | |
| $|\Aut(F/K)|$: | $1$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ 3: $C_3$ x 4 6: $C_6$ x 4 9: $C_3^2$ 12: $A_4$ 18: $C_6 \times C_3$ 24: $A_4\times C_2$ 36: $C_3\times A_4$ 72: 18T25 Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $C_3$
Degree 7: None
Low degree siblings
28T467 x 3, 42T799, 42T800, 42T801, 42T802 x 2, 42T803 x 2, 42T815 x 3Siblings 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, 7, 7 $ | $24$ | $7$ | $( 1, 3, 5, 7, 2, 4, 6)( 8,10,12,14, 9,11,13)(15,21,20,19,18,17,16)$ |
| $ 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $18$ | $7$ | $( 8,10,12,14, 9,11,13)$ |
| $ 7, 7, 7 $ | $72$ | $7$ | $( 1, 3, 5, 7, 2, 4, 6)( 8,12, 9,13,10,14,11)(15,21,20,19,18,17,16)$ |
| $ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ | $36$ | $7$ | $( 1, 6, 4, 2, 7, 5, 3)(15,16,17,18,19,20,21)$ |
| $ 7, 7, 7 $ | $72$ | $7$ | $( 1, 2, 3, 4, 5, 6, 7)( 8,11,14,10,13, 9,12)(15,18,21,17,20,16,19)$ |
| $ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ | $36$ | $7$ | $( 8,10,12,14, 9,11,13)(15,19,16,20,17,21,18)$ |
| $ 7, 7, 7 $ | $24$ | $7$ | $( 1, 3, 5, 7, 2, 4, 6)( 8,12, 9,13,10,14,11)(15,18,21,17,20,16,19)$ |
| $ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ | $36$ | $7$ | $( 8,12, 9,13,10,14,11)(15,19,16,20,17,21,18)$ |
| $ 7, 7, 7 $ | $24$ | $7$ | $( 1, 7, 6, 5, 4, 3, 2)( 8,11,14,10,13, 9,12)(15,16,17,18,19,20,21)$ |
| $ 3, 3, 3, 3, 3, 3, 1, 1, 1 $ | $343$ | $3$ | $( 2, 3, 5)( 4, 7, 6)( 9,10,12)(11,14,13)(16,17,19)(18,21,20)$ |
| $ 3, 3, 3, 3, 3, 3, 1, 1, 1 $ | $343$ | $3$ | $( 2, 5, 3)( 4, 6, 7)( 9,12,10)(11,13,14)(16,19,17)(18,20,21)$ |
| $ 6, 6, 3, 3, 1, 1, 1 $ | $1029$ | $6$ | $( 2, 4, 3, 7, 5, 6)( 9,12,10)(11,13,14)(16,18,17,21,19,20)$ |
| $ 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 $ | $882$ | $14$ | $( 1, 3)( 4, 7)( 5, 6)( 8,10,12,14, 9,11,13)(15,21)(16,20)(17,19)$ |
| $ 6, 6, 3, 3, 1, 1, 1 $ | $1029$ | $6$ | $( 2, 6, 5, 7, 3, 4)( 9,10,12)(11,14,13)(16,20,19,21,17,18)$ |
| $ 3, 3, 3, 3, 3, 3, 3 $ | $196$ | $3$ | $( 1,20,11)( 2,19, 8)( 3,18,12)( 4,17, 9)( 5,16,13)( 6,15,10)( 7,21,14)$ |
| $ 21 $ | $1176$ | $21$ | $( 1,20,13, 5,16, 8, 2,19,10, 6,15,12, 3,18,14, 7,21, 9, 4,17,11)$ |
| $ 21 $ | $1176$ | $21$ | $( 1,18, 9, 7,20,14, 6,15,12, 5,17,10, 4,19, 8, 3,21,13, 2,16,11)$ |
| $ 3, 3, 3, 3, 3, 3, 3 $ | $196$ | $3$ | $( 1,18,11)( 2,16,13)( 3,21, 8)( 4,19,10)( 5,17,12)( 6,15,14)( 7,20, 9)$ |
| $ 3, 3, 3, 3, 3, 3, 3 $ | $196$ | $3$ | $( 1,21,11)( 2,17,12)( 3,20,13)( 4,16,14)( 5,19, 8)( 6,15, 9)( 7,18,10)$ |
| $ 21 $ | $1176$ | $21$ | $( 1,20, 8, 7,17,14, 6,21,13, 5,18,12, 4,15,11, 3,19,10, 2,16, 9)$ |
| $ 3, 3, 3, 3, 3, 3, 3 $ | $196$ | $3$ | $( 1,11,20)( 2, 8,19)( 3,12,18)( 4, 9,17)( 5,13,16)( 6,10,15)( 7,14,21)$ |
| $ 21 $ | $1176$ | $21$ | $( 1,13,16, 5, 8,19, 2,10,15, 6,12,18, 3,14,21, 7, 9,17, 4,11,20)$ |
| $ 3, 3, 3, 3, 3, 3, 3 $ | $196$ | $3$ | $( 1,14,20)( 2, 8,16)( 3, 9,19)( 4,10,15)( 5,11,18)( 6,12,21)( 7,13,17)$ |
| $ 21 $ | $1176$ | $21$ | $( 1, 9,18, 7, 8,15, 6,14,19, 5,13,16, 4,12,20, 3,11,17, 2,10,21)$ |
| $ 21 $ | $1176$ | $21$ | $( 1,13,19, 5,14,18, 2, 8,17, 6, 9,16, 3,10,15, 7,11,21, 4,12,20)$ |
| $ 3, 3, 3, 3, 3, 3, 3 $ | $196$ | $3$ | $( 1,12,20)( 2,14,18)( 3, 9,16)( 4,11,21)( 5,13,19)( 6, 8,17)( 7,10,15)$ |
| $ 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $21$ | $2$ | $(16,21)(17,20)(18,19)$ |
| $ 7, 7, 2, 2, 2, 1 $ | $252$ | $14$ | $( 1, 3, 5, 7, 2, 4, 6)( 8,10,12,14, 9,11,13)(15,21)(16,20)(17,19)$ |
| $ 7, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $126$ | $14$ | $( 8,10,12,14, 9,11,13)(16,21)(17,20)(18,19)$ |
| $ 7, 7, 2, 2, 2, 1 $ | $252$ | $14$ | $( 1, 3, 5, 7, 2, 4, 6)( 8,12, 9,13,10,14,11)(15,21)(16,20)(17,19)$ |
| $ 7, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $126$ | $14$ | $( 1, 6, 4, 2, 7, 5, 3)(15,16)(17,21)(18,20)$ |
| $ 7, 7, 2, 2, 2, 1 $ | $252$ | $14$ | $( 1, 2, 3, 4, 5, 6, 7)( 8,11,14,10,13, 9,12)(15,18)(16,17)(19,21)$ |
| $ 6, 3, 3, 3, 3, 1, 1, 1 $ | $1029$ | $6$ | $( 2, 3, 5)( 4, 7, 6)( 9,10,12)(11,14,13)(16,20,19,21,17,18)$ |
| $ 6, 3, 3, 3, 3, 1, 1, 1 $ | $1029$ | $6$ | $( 2, 5, 3)( 4, 6, 7)( 9,12,10)(11,13,14)(16,18,17,21,19,20)$ |
| $ 6, 6, 6, 1, 1, 1 $ | $343$ | $6$ | $( 2, 4, 3, 7, 5, 6)( 9,11,10,14,12,13)(16,18,17,21,19,20)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1 $ | $343$ | $2$ | $( 2, 7)( 3, 6)( 4, 5)( 9,14)(10,13)(11,12)(16,21)(17,20)(18,19)$ |
| $ 6, 6, 6, 1, 1, 1 $ | $343$ | $6$ | $( 2, 6, 5, 7, 3, 4)( 9,13,12,14,10,11)(16,20,19,21,17,18)$ |
| $ 6, 6, 6, 3 $ | $1372$ | $6$ | $( 1,20, 9, 4,17,11)( 2,19,12, 3,18, 8)( 5,16,14, 7,21,13)( 6,15,10)$ |
| $ 6, 6, 6, 3 $ | $1372$ | $6$ | $( 1,18, 8, 3,21,11)( 2,16,13)( 4,19, 9, 7,20,10)( 5,17,14, 6,15,12)$ |
| $ 6, 6, 6, 3 $ | $1372$ | $6$ | $( 1,21,14, 4,16,11)( 2,17,13, 3,20,12)( 5,19,10, 7,18, 8)( 6,15, 9)$ |
| $ 6, 6, 6, 3 $ | $1372$ | $6$ | $( 1,11,20, 4, 9,17)( 2, 8,19, 3,12,18)( 5,13,16, 7,14,21)( 6,10,15)$ |
| $ 6, 6, 6, 3 $ | $1372$ | $6$ | $( 1,14,20, 7,13,17)( 2, 8,16, 6,12,21)( 3, 9,19, 5,11,18)( 4,10,15)$ |
| $ 6, 6, 6, 3 $ | $1372$ | $6$ | $( 1,13,19, 2, 8,17)( 3,10,15, 7,11,21)( 4,12,20, 6, 9,16)( 5,14,18)$ |
Group invariants
| Order: | $24696=2^{3} \cdot 3^{2} \cdot 7^{3}$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | Data not available |
| Character table: Data not available. |