Group action invariants
| Degree $n$ : | $21$ | |
| Transitive number $t$ : | $43$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,12,17,4,14,15)(2,8,21,3,11,18)(5,10,19,7,9,20)(6,13,16), (1,10,16,5,14,17,2,11,18,6,8,19,3,12,20,7,9,21,4,13,15) | |
| $|\Aut(F/K)|$: | $1$ |
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: $C_3^2$ 18: $C_6 \times C_3$ 42: $F_7$ x 3 126: 21T9 x 3 882: 21T24 x 3 Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $C_3$
Degree 7: None
Low degree siblings
21T43 x 11, 42T467 x 12Siblings 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 $ | $18$ | $7$ | $( 1, 5, 2, 6, 3, 7, 4)(15,20,18,16,21,19,17)$ |
| $ 7, 7, 7 $ | $6$ | $7$ | $( 1, 6, 4, 2, 7, 5, 3)( 8,13,11, 9,14,12,10)(15,18,21,17,20,16,19)$ |
| $ 7, 7, 7 $ | $18$ | $7$ | $( 1, 2, 3, 4, 5, 6, 7)( 8,13,11, 9,14,12,10)(15,20,18,16,21,19,17)$ |
| $ 7, 7, 7 $ | $6$ | $7$ | $( 1, 7, 6, 5, 4, 3, 2)( 8,13,11, 9,14,12,10)(15,21,20,19,18,17,16)$ |
| $ 3, 3, 3, 3, 3, 3, 3 $ | $49$ | $3$ | $( 1,17,14)( 2,21,11)( 3,18, 8)( 4,15,12)( 5,19, 9)( 6,16,13)( 7,20,10)$ |
| $ 3, 3, 3, 3, 3, 3, 3 $ | $49$ | $3$ | $( 1,14,17)( 2,11,21)( 3, 8,18)( 4,12,15)( 5, 9,19)( 6,13,16)( 7,10,20)$ |
| $ 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $18$ | $7$ | $(15,20,18,16,21,19,17)$ |
| $ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ | $18$ | $7$ | $( 1, 5, 2, 6, 3, 7, 4)(15,18,21,17,20,16,19)$ |
| $ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ | $18$ | $7$ | $( 1, 6, 4, 2, 7, 5, 3)(15,21,20,19,18,17,16)$ |
| $ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ | $18$ | $7$ | $( 1, 2, 3, 4, 5, 6, 7)(15,16,17,18,19,20,21)$ |
| $ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ | $18$ | $7$ | $( 1, 3, 5, 7, 2, 4, 6)(15,19,16,20,17,21,18)$ |
| $ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ | $18$ | $7$ | $( 1, 7, 6, 5, 4, 3, 2)(15,17,19,21,16,18,20)$ |
| $ 7, 7, 7 $ | $18$ | $7$ | $( 1, 5, 2, 6, 3, 7, 4)( 8,13,11, 9,14,12,10)(15,20,18,16,21,19,17)$ |
| $ 7, 7, 7 $ | $18$ | $7$ | $( 1, 6, 4, 2, 7, 5, 3)( 8,13,11, 9,14,12,10)(15,16,17,18,19,20,21)$ |
| $ 7, 7, 7 $ | $18$ | $7$ | $( 1, 2, 3, 4, 5, 6, 7)( 8,13,11, 9,14,12,10)(15,18,21,17,20,16,19)$ |
| $ 7, 7, 7 $ | $18$ | $7$ | $( 1, 3, 5, 7, 2, 4, 6)( 8,13,11, 9,14,12,10)(15,21,20,19,18,17,16)$ |
| $ 7, 7, 7 $ | $18$ | $7$ | $( 1, 7, 6, 5, 4, 3, 2)( 8,13,11, 9,14,12,10)(15,19,16,20,17,21,18)$ |
| $ 7, 7, 7 $ | $18$ | $7$ | $( 1, 5, 2, 6, 3, 7, 4)( 8, 9,10,11,12,13,14)(15,17,19,21,16,18,20)$ |
| $ 7, 7, 7 $ | $18$ | $7$ | $( 1, 3, 5, 7, 2, 4, 6)( 8, 9,10,11,12,13,14)(15,18,21,17,20,16,19)$ |
| $ 7, 7, 7 $ | $18$ | $7$ | $( 1, 6, 4, 2, 7, 5, 3)( 8,11,14,10,13, 9,12)(15,18,21,17,20,16,19)$ |
| $ 7, 7, 7 $ | $18$ | $7$ | $( 1, 6, 4, 2, 7, 5, 3)( 8,10,12,14, 9,11,13)(15,19,16,20,17,21,18)$ |
| $ 7, 7, 7 $ | $18$ | $7$ | $( 1, 2, 3, 4, 5, 6, 7)( 8,10,12,14, 9,11,13)(15,21,20,19,18,17,16)$ |
| $ 7, 7, 7 $ | $6$ | $7$ | $( 1, 2, 3, 4, 5, 6, 7)( 8,12, 9,13,10,14,11)(15,19,16,20,17,21,18)$ |
| $ 21 $ | $294$ | $21$ | $( 1,17,12, 4,15,10, 7,20, 8, 3,18,13, 6,16,11, 2,21, 9, 5,19,14)$ |
| $ 21 $ | $294$ | $21$ | $( 1,14,17, 4,12,15, 7,10,20, 3, 8,18, 6,13,16, 2,11,21, 5, 9,19)$ |
| $ 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 $ | $343$ | $6$ | $( 1,17,10, 6,18, 9)( 2,20,14, 5,15,12)( 3,16,11, 4,19, 8)( 7,21,13)$ |
| $ 6, 6, 6, 3 $ | $343$ | $6$ | $( 1,14,19, 3,13,20)( 2,10,16)( 4, 9,17, 7,11,15)( 5,12,21, 6, 8,18)$ |
| $ 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 $ | $294$ | $21$ | $( 1,17, 9, 7,16,14, 6,15,12, 5,21,10, 4,20, 8, 3,19,13, 2,18,11)$ |
| $ 3, 3, 3, 3, 3, 3, 3 $ | $49$ | $3$ | $( 1,16,14)( 2,17, 9)( 3,18,11)( 4,19,13)( 5,20, 8)( 6,21,10)( 7,15,12)$ |
| $ 3, 3, 3, 3, 3, 3, 3 $ | $49$ | $3$ | $( 1,14,16)( 2, 8,18)( 3, 9,20)( 4,10,15)( 5,11,17)( 6,12,19)( 7,13,21)$ |
| $ 21 $ | $294$ | $21$ | $( 1,14,21, 4,10,20, 7,13,19, 3, 9,18, 6,12,17, 2, 8,16, 5,11,15)$ |
| $ 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 $ | $343$ | $6$ | $( 1,17, 8, 3,15,12)( 2,16,10)( 4,21,14, 7,18,13)( 5,20, 9, 6,19,11)$ |
| $ 6, 6, 6, 3 $ | $343$ | $6$ | $( 1,14,20, 5,10,21)( 2,13,15, 4,11,19)( 3,12,17)( 6, 9,16, 7, 8,18)$ |
| $ 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, 3 $ | $49$ | $3$ | $( 1,17,13)( 2,19,14)( 3,21, 8)( 4,16, 9)( 5,18,10)( 6,20,11)( 7,15,12)$ |
| $ 21 $ | $294$ | $21$ | $( 1,15,12, 4,21, 8, 7,20,11, 3,19,14, 6,18,10, 2,17,13, 5,16, 9)$ |
| $ 21 $ | $294$ | $21$ | $( 1,14,21, 3,11,16, 5, 8,18, 7,12,20, 2, 9,15, 4,13,17, 6,10,19)$ |
| $ 3, 3, 3, 3, 3, 3, 3 $ | $49$ | $3$ | $( 1,14,16)( 2, 9,17)( 3,11,18)( 4,13,19)( 5, 8,20)( 6,10,21)( 7,12,15)$ |
| $ 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 $ | $343$ | $6$ | $( 1,17,11, 7,19,10)( 2,15,12, 6,21, 9)( 3,20,13, 5,16, 8)( 4,18,14)$ |
| $ 6, 6, 6, 3 $ | $343$ | $6$ | $( 1,14,15, 4, 8,18)( 2,12,16, 3,10,17)( 5,13,19, 7, 9,21)( 6,11,20)$ |
Group invariants
| Order: | $6174=2 \cdot 3^{2} \cdot 7^{3}$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | Data not available |
| Character table: Data not available. |