Group action invariants
| Degree $n$ : | $21$ | |
| Transitive number $t$ : | $55$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,20,13,3,21,14,5,15,8,7,16,9,2,17,10,4,18,11,6,19,12), (1,2)(3,7)(4,6)(9,14)(10,13)(11,12)(15,19)(16,18)(20,21), (1,17,4,21,2,16)(3,15,5,20,6,19)(7,18)(8,13,10,9,11,14) | |
| $|\Aut(F/K)|$: | $1$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ x 3 3: $C_3$ 4: $C_2^2$ 6: $S_3$, $C_6$ x 3 12: $D_{6}$, $C_6\times C_2$ 18: $S_3\times C_3$ 36: $C_6\times S_3$ 42: $F_7$ 84: $F_7 \times C_2$ 252: 21T15 1764: 14T37 Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $S_3$
Degree 7: None
Low degree siblings
21T55 x 5, 42T633 x 6, 42T634 x 6, 42T635 x 6, 42T638 x 3, 42T639 x 2Siblings 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 $ | $6$ | $7$ | $( 1, 3, 5, 7, 2, 4, 6)( 8, 9,10,11,12,13,14)(15,16,17,18,19,20,21)$ |
| $ 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 $ | $18$ | $7$ | $( 1, 3, 5, 7, 2, 4, 6)( 8,11,14,10,13, 9,12)(15,16,17,18,19,20,21)$ |
| $ 7, 7, 7 $ | $18$ | $7$ | $( 1, 7, 6, 5, 4, 3, 2)( 8,13,11, 9,14,12,10)(15,18,21,17,20,16,19)$ |
| $ 7, 7, 7 $ | $18$ | $7$ | $( 1, 5, 2, 6, 3, 7, 4)( 8,12, 9,13,10,14,11)(15,17,19,21,16,18,20)$ |
| $ 7, 7, 7 $ | $18$ | $7$ | $( 1, 6, 4, 2, 7, 5, 3)( 8, 9,10,11,12,13,14)(15,21,20,19,18,17,16)$ |
| $ 7, 7, 7 $ | $18$ | $7$ | $( 1, 2, 3, 4, 5, 6, 7)( 8,14,13,12,11,10, 9)(15,19,16,20,17,21,18)$ |
| $ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ | $18$ | $7$ | $( 1, 4, 7, 3, 6, 2, 5)(15,20,18,16,21,19,17)$ |
| $ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ | $18$ | $7$ | $( 8,10,12,14, 9,11,13)(15,20,18,16,21,19,17)$ |
| $ 7, 7, 7 $ | $36$ | $7$ | $( 1, 3, 5, 7, 2, 4, 6)( 8,11,14,10,13, 9,12)(15,21,20,19,18,17,16)$ |
| $ 7, 7, 7 $ | $36$ | $7$ | $( 1, 7, 6, 5, 4, 3, 2)( 8,13,11, 9,14,12,10)(15,16,17,18,19,20,21)$ |
| $ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ | $36$ | $7$ | $( 1, 5, 2, 6, 3, 7, 4)( 8,12, 9,13,10,14,11)$ |
| $ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ | $36$ | $7$ | $( 8,12, 9,13,10,14,11)(15,20,18,16,21,19,17)$ |
| $ 7, 7, 7 $ | $36$ | $7$ | $( 1, 3, 5, 7, 2, 4, 6)( 8,13,11, 9,14,12,10)(15,21,20,19,18,17,16)$ |
| $ 7, 7, 7 $ | $12$ | $7$ | $( 1, 2, 3, 4, 5, 6, 7)( 8, 9,10,11,12,13,14)(15,17,19,21,16,18,20)$ |
| $ 21 $ | $588$ | $21$ | $( 1,20,13, 3,21,14, 5,15, 8, 7,16, 9, 2,17,10, 4,18,11, 6,19,12)$ |
| $ 3, 3, 3, 3, 3, 3, 3 $ | $98$ | $3$ | $( 1,15,10)( 2,19,14)( 3,16,11)( 4,20, 8)( 5,17,12)( 6,21, 9)( 7,18,13)$ |
| $ 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, 3 $ | $686$ | $6$ | $( 1,20,10, 3,19,11)( 2,16,14)( 4,15, 8, 7,17,13)( 5,18,12, 6,21, 9)$ |
| $ 3, 3, 3, 3, 3, 3, 1, 1, 1 $ | $343$ | $3$ | $( 2, 5, 3)( 4, 6, 7)( 9,12,10)(11,13,14)(16,19,17)(18,20,21)$ |
| $ 21 $ | $588$ | $21$ | $( 1,20,14, 6,16,12, 4,19,10, 2,15, 8, 7,18,13, 5,21,11, 3,17, 9)$ |
| $ 3, 3, 3, 3, 3, 3, 3 $ | $98$ | $3$ | $( 1,20, 9)( 2,15,10)( 3,17,11)( 4,19,12)( 5,21,13)( 6,16,14)( 7,18, 8)$ |
| $ 6, 6, 6, 1, 1, 1 $ | $343$ | $6$ | $( 2, 4, 3, 7, 5, 6)( 9,11,10,14,12,13)(16,18,17,21,19,20)$ |
| $ 6, 6, 6, 3 $ | $686$ | $6$ | $( 1,20, 9, 6,17,14)( 2,18,10, 5,19,13)( 3,16,11, 4,21,12)( 7,15, 8)$ |
| $ 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 $ | $588$ | $21$ | $( 1,20,11, 5,17,12, 2,21,13, 6,18,14, 3,15, 8, 7,19, 9, 4,16,10)$ |
| $ 3, 3, 3, 3, 3, 3, 3 $ | $98$ | $3$ | $( 1,20,10)( 2,21,12)( 3,15,14)( 4,16, 9)( 5,17,11)( 6,18,13)( 7,19, 8)$ |
| $ 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 $ | $686$ | $6$ | $( 1,20,12, 5,16,13)( 2,19,14, 4,17,11)( 3,18, 9)( 6,15, 8, 7,21,10)$ |
| $ 6, 6, 6, 2, 1 $ | $1029$ | $6$ | $( 2, 6, 5, 7, 3, 4)( 8,16)( 9,21,12,15,10,19)(11,17,13,20,14,18)$ |
| $ 6, 6, 3, 3, 2, 1 $ | $1029$ | $6$ | $( 2, 3, 5)( 4, 7, 6)( 8,21,11,20, 9,16)(10,18,12,15,13,17)(14,19)$ |
| $ 14, 2, 2, 2, 1 $ | $882$ | $14$ | $( 2, 7)( 3, 6)( 4, 5)( 8,19,12,15, 9,18,13,21,10,17,14,20,11,16)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ | $147$ | $2$ | $( 2, 7)( 3, 6)( 4, 5)( 8,19)( 9,18)(10,17)(11,16)(12,15)(13,21)(14,20)$ |
| $ 14, 1, 1, 1, 1, 1, 1, 1 $ | $126$ | $14$ | $( 8,18,10,20,12,15,14,17, 9,19,11,21,13,16)$ |
| $ 14, 7 $ | $126$ | $14$ | $( 1, 3, 5, 7, 2, 4, 6)( 8,19,12,16, 9,20,13,17,10,21,14,18,11,15)$ |
| $ 14, 7 $ | $126$ | $14$ | $( 1, 7, 6, 5, 4, 3, 2)( 8,21, 9,15,10,16,11,17,12,18,13,19,14,20)$ |
| $ 14, 7 $ | $126$ | $14$ | $( 1, 5, 2, 6, 3, 7, 4)( 8,20,14,19,13,18,12,17,11,16,10,15, 9,21)$ |
| $ 7, 2, 2, 2, 2, 2, 2, 2 $ | $126$ | $14$ | $( 1, 6, 4, 2, 7, 5, 3)( 8,17)( 9,18)(10,19)(11,20)(12,21)(13,15)(14,16)$ |
| $ 14, 7 $ | $126$ | $14$ | $( 1, 2, 3, 4, 5, 6, 7)( 8,15,11,18,14,21,10,17,13,20, 9,16,12,19)$ |
| $ 14, 7 $ | $126$ | $14$ | $( 1, 4, 7, 3, 6, 2, 5)( 8,16,13,21,11,19, 9,17,14,15,12,20,10,18)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1 $ | $21$ | $2$ | $( 8,18)( 9,19)(10,20)(11,21)(12,15)(13,16)(14,17)$ |
| $ 14, 7 $ | $126$ | $14$ | $( 1, 3, 5, 7, 2, 4, 6)( 8,19,10,21,12,16,14,18, 9,20,11,15,13,17)$ |
| $ 6, 6, 6, 2, 1 $ | $1029$ | $6$ | $( 2, 4, 3, 7, 5, 6)( 8,17,11,19,10,16)( 9,20,13,18,14,21)(12,15)$ |
| $ 6, 6, 3, 3, 2, 1 $ | $1029$ | $6$ | $( 2, 5, 3)( 4, 6, 7)( 8,20,10,21,14,16)( 9,17,12,15,11,18)(13,19)$ |
Group invariants
| Order: | $12348=2^{2} \cdot 3^{2} \cdot 7^{3}$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | Data not available |
| Character table: Data not available. |