Group action invariants
| Degree $n$ : | $21$ | |
| Transitive number $t$ : | $69$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,11,4,9,5,13,3,12,7,14,6,10)(2,8)(15,19,17,18,21,16), (1,16,13,7,19,10,6,15,14,5,18,11,4,21,8,3,17,12,2,20,9) | |
| $|\Aut(F/K)|$: | $1$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ 3: $C_3$ 6: $S_3$, $C_6$ 18: $S_3\times C_3$ 24: $S_4$ 72: 12T45 Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $S_3$
Degree 7: None
Low degree siblings
28T465, 42T796, 42T797, 42T798, 42T818Siblings 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, 1, 1, 1, 1, 1, 1, 1 $ | $36$ | $7$ | $( 1, 6, 4, 2, 7, 5, 3)( 8,13,11, 9,14,12,10)$ |
| $ 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $18$ | $7$ | $( 8,13,11, 9,14,12,10)$ |
| $ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ | $72$ | $7$ | $( 1, 6, 4, 2, 7, 5, 3)( 8,11,14,10,13, 9,12)$ |
| $ 7, 7, 7 $ | $12$ | $7$ | $( 1, 6, 4, 2, 7, 5, 3)( 8,13,11, 9,14,12,10)(15,19,16,20,17,21,18)$ |
| $ 7, 7, 7 $ | $36$ | $7$ | $( 1, 2, 3, 4, 5, 6, 7)( 8, 9,10,11,12,13,14)(15,19,16,20,17,21,18)$ |
| $ 7, 7, 7 $ | $36$ | $7$ | $( 1, 4, 7, 3, 6, 2, 5)( 8,11,14,10,13, 9,12)(15,19,16,20,17,21,18)$ |
| $ 7, 7, 7 $ | $36$ | $7$ | $( 1, 2, 3, 4, 5, 6, 7)( 8,14,13,12,11,10, 9)(15,19,16,20,17,21,18)$ |
| $ 7, 7, 7 $ | $24$ | $7$ | $( 1, 4, 7, 3, 6, 2, 5)( 8, 9,10,11,12,13,14)(15,19,16,20,17,21,18)$ |
| $ 7, 7, 7 $ | $36$ | $7$ | $( 1, 4, 7, 3, 6, 2, 5)( 8,12, 9,13,10,14,11)(15,19,16,20,17,21,18)$ |
| $ 7, 7, 7 $ | $24$ | $7$ | $( 1, 2, 3, 4, 5, 6, 7)( 8,12, 9,13,10,14,11)(15,19,16,20,17,21,18)$ |
| $ 7, 7, 7 $ | $12$ | $7$ | $( 1, 3, 5, 7, 2, 4, 6)( 8,13,11, 9,14,12,10)(15,19,16,20,17,21,18)$ |
| $ 3, 3, 3, 3, 3, 3, 1, 1, 1 $ | $343$ | $3$ | $( 2, 5, 3)( 4, 6, 7)( 9,12,10)(11,13,14)(16,19,17)(18,20,21)$ |
| $ 3, 3, 3, 3, 3, 3, 1, 1, 1 $ | $343$ | $3$ | $( 2, 3, 5)( 4, 7, 6)( 9,10,12)(11,14,13)(16,17,19)(18,21,20)$ |
| $ 6, 6, 3, 3, 1, 1, 1 $ | $1029$ | $6$ | $( 2, 6, 5, 7, 3, 4)( 9,13,12,14,10,11)(16,17,19)(18,21,20)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $147$ | $2$ | $( 2, 7)( 3, 6)( 4, 5)( 9,14)(10,13)(11,12)$ |
| $ 7, 2, 2, 2, 2, 2, 2, 1, 1 $ | $882$ | $14$ | $( 2, 7)( 3, 6)( 4, 5)( 9,14)(10,13)(11,12)(15,19,16,20,17,21,18)$ |
| $ 6, 6, 3, 3, 1, 1, 1 $ | $1029$ | $6$ | $( 2, 4, 3, 7, 5, 6)( 9,11,10,14,12,13)(16,19,17)(18,20,21)$ |
| $ 21 $ | $1176$ | $21$ | $( 1,14,19, 3,12,15, 5,10,18, 7, 8,21, 2,13,17, 4,11,20, 6, 9,16)$ |
| $ 21 $ | $1176$ | $21$ | $( 1,10,18, 7,11,20, 6,12,15, 5,13,17, 4,14,19, 3, 8,21, 2, 9,16)$ |
| $ 3, 3, 3, 3, 3, 3, 3 $ | $392$ | $3$ | $( 1, 9,16)( 2, 8,21)( 3,14,19)( 4,13,17)( 5,12,15)( 6,11,20)( 7,10,18)$ |
| $ 21 $ | $1176$ | $21$ | $( 1,11,21, 5, 9,19, 2,14,17, 6,12,15, 3,10,20, 7, 8,18, 4,13,16)$ |
| $ 21 $ | $1176$ | $21$ | $( 1, 9,19, 7,13,16, 6,10,20, 5,14,17, 4,11,21, 3, 8,18, 2,12,15)$ |
| $ 3, 3, 3, 3, 3, 3, 3 $ | $392$ | $3$ | $( 1, 8,18)( 2,11,21)( 3,14,17)( 4,10,20)( 5,13,16)( 6, 9,19)( 7,12,15)$ |
| $ 21 $ | $1176$ | $21$ | $( 1,13,19, 5,12,15, 2,11,18, 6,10,21, 3, 9,17, 7, 8,20, 4,14,16)$ |
| $ 21 $ | $1176$ | $21$ | $( 1,11,18, 4,12,15, 7,13,19, 3,14,16, 6, 8,20, 2, 9,17, 5,10,21)$ |
| $ 3, 3, 3, 3, 3, 3, 3 $ | $392$ | $3$ | $( 1,12,15)( 2,10,21)( 3, 8,20)( 4,13,19)( 5,11,18)( 6, 9,17)( 7,14,16)$ |
| $ 12, 6, 2, 1 $ | $2058$ | $12$ | $( 2, 4, 3, 7, 5, 6)( 8,17,10,19,13,15,14,16,12,21, 9,18)(11,20)$ |
| $ 12, 6, 2, 1 $ | $2058$ | $12$ | $( 2, 6, 5, 7, 3, 4)( 8,16,10,17, 9,20,13,15,11,21,12,18)(14,19)$ |
| $ 4, 4, 4, 2, 2, 2, 2, 1 $ | $2058$ | $4$ | $( 2, 7)( 3, 6)( 4, 5)( 8,19,11,18)( 9,21,10,16)(12,20,14,17)(13,15)$ |
| $ 14, 1, 1, 1, 1, 1, 1, 1 $ | $252$ | $14$ | $( 8,19,12,20, 9,21,13,15,10,16,14,17,11,18)$ |
| $ 14, 7 $ | $252$ | $14$ | $( 1, 6, 4, 2, 7, 5, 3)( 8,19,10,16,12,20,14,17, 9,21,11,18,13,15)$ |
| $ 14, 7 $ | $252$ | $14$ | $( 1, 2, 3, 4, 5, 6, 7)( 8,19,13,15,11,18, 9,21,14,17,12,20,10,16)$ |
| $ 7, 2, 2, 2, 2, 2, 2, 2 $ | $126$ | $14$ | $( 1, 4, 7, 3, 6, 2, 5)( 8,19)( 9,21)(10,16)(11,18)(12,20)(13,15)(14,17)$ |
| $ 14, 7 $ | $252$ | $14$ | $( 1, 3, 5, 7, 2, 4, 6)( 8,19,14,17,13,15,12,20,11,18,10,16, 9,21)$ |
| $ 14, 7 $ | $252$ | $14$ | $( 1, 7, 6, 5, 4, 3, 2)( 8,19,11,18,14,17,10,16,13,15, 9,21,12,20)$ |
| $ 14, 7 $ | $252$ | $14$ | $( 1, 6, 4, 2, 7, 5, 3)( 8,19,11,18,14,17,10,16,13,15, 9,21,12,20)$ |
| $ 14, 7 $ | $252$ | $14$ | $( 1, 2, 3, 4, 5, 6, 7)( 8,19,14,17,13,15,12,20,11,18,10,16, 9,21)$ |
| $ 7, 2, 2, 2, 2, 2, 2, 2 $ | $126$ | $14$ | $( 1, 3, 5, 7, 2, 4, 6)( 8,19)( 9,21)(10,16)(11,18)(12,20)(13,15)(14,17)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1 $ | $42$ | $2$ | $( 8,19)( 9,21)(10,16)(11,18)(12,20)(13,15)(14,17)$ |
| $ 6, 6, 3, 3, 2, 1 $ | $2058$ | $6$ | $( 2, 5, 3)( 4, 6, 7)( 8,17,13,15, 9,18)(10,19)(11,20,12,21,14,16)$ |
| $ 6, 6, 3, 3, 2, 1 $ | $2058$ | $6$ | $( 2, 3, 5)( 4, 7, 6)( 8,16,13,15,12,18)( 9,20,10,17,14,19)(11,21)$ |
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. |