Group action invariants
| Degree $n$ : | $21$ | |
| Transitive number $t$ : | $72$ | |
| Parity: | $1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,17,8,5,20,9,2,16,10,6,19,11,3,15,12,7,18,13,4,21,14), (1,16,4,19,5,20,3,18,7,15,6,21)(2,17)(8,12,14)(10,13,11) | |
| $|\Aut(F/K)|$: | $1$ |
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
28T466, 42T811, 42T812, 42T813, 42T817Siblings 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 $ | $24$ | $7$ | $( 1, 5, 2, 6, 3, 7, 4)( 8, 9,10,11,12,13,14)(15,18,21,17,20,16,19)$ |
| $ 7, 7, 7 $ | $24$ | $7$ | $( 1, 6, 4, 2, 7, 5, 3)( 8,11,14,10,13, 9,12)(15,17,19,21,16,18,20)$ |
| $ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ | $72$ | $7$ | $( 8,10,12,14, 9,11,13)(15,20,18,16,21,19,17)$ |
| $ 7, 7, 7 $ | $24$ | $7$ | $( 1, 5, 2, 6, 3, 7, 4)( 8,11,14,10,13, 9,12)(15,16,17,18,19,20,21)$ |
| $ 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, 7, 7 $ | $72$ | $7$ | $( 1, 4, 7, 3, 6, 2, 5)( 8, 9,10,11,12,13,14)(15,17,19,21,16,18,20)$ |
| $ 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $18$ | $7$ | $(15,18,21,17,20,16,19)$ |
| $ 7, 7, 7 $ | $72$ | $7$ | $( 1, 5, 2, 6, 3, 7, 4)( 8, 9,10,11,12,13,14)(15,21,20,19,18,17,16)$ |
| $ 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 $ | $882$ | $14$ | $( 1, 5)( 2, 4)( 6, 7)( 8, 9)(10,14)(11,13)(15,18,21,17,20,16,19)$ |
| $ 21 $ | $1176$ | $21$ | $( 1,17, 8, 5,20, 9, 2,16,10, 6,19,11, 3,15,12, 7,18,13, 4,21,14)$ |
| $ 21 $ | $1176$ | $21$ | $( 1,19,14, 6,21,10, 4,16,13, 2,18, 9, 7,20,12, 5,15, 8, 3,17,11)$ |
| $ 3, 3, 3, 3, 3, 3, 3 $ | $392$ | $3$ | $( 1,16,12)( 2,15,14)( 3,21, 9)( 4,20,11)( 5,19,13)( 6,18, 8)( 7,17,10)$ |
| $ 21 $ | $1176$ | $21$ | $( 1, 8,20, 2,10,19, 3,12,18, 4,14,17, 5, 9,16, 6,11,15, 7,13,21)$ |
| $ 21 $ | $1176$ | $21$ | $( 1, 9,19, 7,14,20, 6,12,21, 5,10,15, 4, 8,16, 3,13,17, 2,11,18)$ |
| $ 3, 3, 3, 3, 3, 3, 3 $ | $392$ | $3$ | $( 1,12,16)( 2,14,15)( 3, 9,21)( 4,11,20)( 5,13,19)( 6, 8,18)( 7,10,17)$ |
| $ 6, 6, 3, 3, 1, 1, 1 $ | $1029$ | $6$ | $( 2, 3, 5)( 4, 7, 6)( 9,13,12,14,10,11)(16,20,19,21,17,18)$ |
| $ 3, 3, 3, 3, 3, 3, 1, 1, 1 $ | $343$ | $3$ | $( 2, 3, 5)( 4, 7, 6)( 9,10,12)(11,14,13)(16,17,19)(18,21,20)$ |
| $ 21 $ | $1176$ | $21$ | $( 1,17,13, 7,19,14, 6,21, 8, 5,16, 9, 4,18,10, 3,20,11, 2,15,12)$ |
| $ 21 $ | $1176$ | $21$ | $( 1,20,12, 5,19, 8, 2,18,11, 6,17,14, 3,16,10, 7,15,13, 4,21, 9)$ |
| $ 3, 3, 3, 3, 3, 3, 3 $ | $392$ | $3$ | $( 1,19,10)( 2,17, 9)( 3,15, 8)( 4,20,14)( 5,18,13)( 6,16,12)( 7,21,11)$ |
| $ 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 $ | $1029$ | $6$ | $( 2, 4, 3, 7, 5, 6)( 9,11,10,14,12,13)(16,19,17)(18,20,21)$ |
| $ 14, 2, 2, 2, 1 $ | $1764$ | $14$ | $( 2, 7)( 3, 6)( 4, 5)( 8,16,11,17,14,18,10,19,13,20, 9,21,12,15)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ | $294$ | $2$ | $( 1, 4)( 2, 3)( 5, 7)( 8,20)( 9,18)(10,16)(11,21)(12,19)(13,17)(14,15)$ |
| $ 4, 4, 4, 2, 1, 1, 1, 1, 1, 1, 1 $ | $294$ | $4$ | $( 8,16,12,15)( 9,21,11,17)(10,19)(13,20,14,18)$ |
| $ 7, 4, 4, 4, 2 $ | $882$ | $28$ | $( 1, 5, 2, 6, 3, 7, 4)( 8,19,11,20)( 9,17,10,15)(12,18,14,21)(13,16)$ |
| $ 7, 4, 4, 4, 2 $ | $882$ | $28$ | $( 1, 6, 4, 2, 7, 5, 3)( 8,18, 9,16)(10,21,14,20)(11,19,13,15)(12,17)$ |
| $ 12, 3, 3, 2, 1 $ | $2058$ | $12$ | $( 1,17)( 2,18, 6,15, 5,21, 7,16, 3,19, 4,20)( 8,10,11)( 9,14,13)$ |
| $ 6, 6, 6, 2, 1 $ | $2058$ | $6$ | $( 1,17)( 2,18, 3,19, 5,21)( 4,20, 7,16, 6,15)( 8,13,14,10,12,11)$ |
| $ 12, 3, 3, 2, 1 $ | $2058$ | $12$ | $( 1, 8, 6,12, 7,10, 3,11, 5,14, 4, 9)( 2,13)(15,20,16)(18,19,21)$ |
| $ 6, 6, 6, 2, 1 $ | $2058$ | $6$ | $( 1, 8, 6,11, 5, 9)( 2,10, 3,12, 7,13)( 4,14)(15,17,20,21,19,16)$ |
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. |