Group action invariants
| Degree $n$ : | $21$ | |
| Transitive number $t$ : | $53$ | |
| Parity: | $1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,8,17,6,14,15,4,13,20,2,12,18,7,11,16,5,10,21,3,9,19), (2,6,5,7,3,4)(8,14,9,12,13,11)(16,17,19)(18,21,20) | |
| $|\Aut(F/K)|$: | $1$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 3: $C_3$ x 4 9: $C_3^2$ 12: $A_4$ 36: $C_3\times A_4$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $C_3$
Degree 7: None
Low degree siblings
28T394 x 3, 42T628, 42T629 x 2, 42T644 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 $ | $12$ | $7$ | $( 1, 6, 4, 2, 7, 5, 3)( 8,14,13,12,11,10, 9)(15,20,18,16,21,19,17)$ |
| $ 7, 7, 7 $ | $12$ | $7$ | $( 1, 2, 3, 4, 5, 6, 7)( 8,12, 9,13,10,14,11)(15,16,17,18,19,20,21)$ |
| $ 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $18$ | $7$ | $(15,21,20,19,18,17,16)$ |
| $ 7, 7, 7 $ | $36$ | $7$ | $( 1, 6, 4, 2, 7, 5, 3)( 8,14,13,12,11,10, 9)(15,19,16,20,17,21,18)$ |
| $ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ | $36$ | $7$ | $( 1, 2, 3, 4, 5, 6, 7)( 8,12, 9,13,10,14,11)$ |
| $ 7, 7, 7 $ | $36$ | $7$ | $( 1, 4, 7, 3, 6, 2, 5)( 8,13,11, 9,14,12,10)(15,17,19,21,16,18,20)$ |
| $ 7, 7, 7 $ | $36$ | $7$ | $( 1, 7, 6, 5, 4, 3, 2)( 8,11,14,10,13, 9,12)(15,20,18,16,21,19,17)$ |
| $ 7, 7, 7 $ | $36$ | $7$ | $( 1, 3, 5, 7, 2, 4, 6)( 8, 9,10,11,12,13,14)(15,21,20,19,18,17,16)$ |
| $ 7, 7, 7 $ | $12$ | $7$ | $( 1, 6, 4, 2, 7, 5, 3)( 8,11,14,10,13, 9,12)(15,19,16,20,17,21,18)$ |
| $ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ | $36$ | $7$ | $( 1, 2, 3, 4, 5, 6, 7)( 8, 9,10,11,12,13,14)$ |
| $ 7, 7, 7 $ | $12$ | $7$ | $( 1, 3, 5, 7, 2, 4, 6)( 8,13,11, 9,14,12,10)(15,16,17,18,19,20,21)$ |
| $ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ | $36$ | $7$ | $( 1, 7, 6, 5, 4, 3, 2)(15,20,18,16,21,19,17)$ |
| $ 7, 7, 7 $ | $12$ | $7$ | $( 1, 3, 5, 7, 2, 4, 6)( 8,13,11, 9,14,12,10)(15,21,20,19,18,17,16)$ |
| $ 7, 7, 7 $ | $12$ | $7$ | $( 1, 5, 2, 6, 3, 7, 4)( 8,14,13,12,11,10, 9)(15,16,17,18,19,20,21)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $147$ | $2$ | $( 9,14)(10,13)(11,12)(16,21)(17,20)(18,19)$ |
| $ 7, 2, 2, 2, 2, 2, 2, 1, 1 $ | $882$ | $14$ | $( 1, 6, 4, 2, 7, 5, 3)( 8,14)( 9,13)(10,12)(15,20)(16,19)(17,18)$ |
| $ 21 $ | $588$ | $21$ | $( 1, 8,17, 6,14,15, 4,13,20, 2,12,18, 7,11,16, 5,10,21, 3, 9,19)$ |
| $ 21 $ | $588$ | $21$ | $( 1,12,19, 2, 9,20, 3,13,21, 4,10,15, 5,14,16, 6,11,17, 7, 8,18)$ |
| $ 3, 3, 3, 3, 3, 3, 3 $ | $196$ | $3$ | $( 1,13,16)( 2,10,17)( 3,14,18)( 4,11,19)( 5, 8,20)( 6,12,21)( 7, 9,15)$ |
| $ 21 $ | $588$ | $21$ | $( 1,17,14, 4,20,12, 7,16,10, 3,19, 8, 6,15,13, 2,18,11, 5,21, 9)$ |
| $ 21 $ | $588$ | $21$ | $( 1,15,12, 5,19,14, 2,16, 9, 6,20,11, 3,17,13, 7,21, 8, 4,18,10)$ |
| $ 3, 3, 3, 3, 3, 3, 3 $ | $196$ | $3$ | $( 1,16,13)( 2,17,10)( 3,18,14)( 4,19,11)( 5,20, 8)( 6,21,12)( 7,15, 9)$ |
| $ 3, 3, 3, 3, 3, 3, 1, 1, 1 $ | $343$ | $3$ | $( 2, 5, 3)( 4, 6, 7)( 9,12,10)(11,13,14)(16,19,17)(18,20,21)$ |
| $ 6, 6, 3, 3, 1, 1, 1 $ | $1029$ | $6$ | $( 2, 5, 3)( 4, 6, 7)( 9,11,10,14,12,13)(16,18,17,21,19,20)$ |
| $ 21 $ | $588$ | $21$ | $( 1, 8,17, 5, 9,18, 2,10,19, 6,11,20, 3,12,21, 7,13,15, 4,14,16)$ |
| $ 3, 3, 3, 3, 3, 3, 3 $ | $196$ | $3$ | $( 1, 8,16)( 2,10,18)( 3,12,20)( 4,14,15)( 5, 9,17)( 6,11,19)( 7,13,21)$ |
| $ 21 $ | $588$ | $21$ | $( 1, 8,21, 7,13,19, 6,11,17, 5, 9,15, 4,14,20, 3,12,18, 2,10,16)$ |
| $ 3, 3, 3, 3, 3, 3, 3 $ | $196$ | $3$ | $( 1,17,10)( 2,21,11)( 3,18,12)( 4,15,13)( 5,19,14)( 6,16, 8)( 7,20, 9)$ |
| $ 21 $ | $588$ | $21$ | $( 1,16, 8, 6,15,13, 4,21,11, 2,20, 9, 7,19,14, 5,18,12, 3,17,10)$ |
| $ 21 $ | $588$ | $21$ | $( 1,21,11, 2,18,12, 3,15,13, 4,19,14, 5,16, 8, 6,20, 9, 7,17,10)$ |
| $ 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 $ | $1029$ | $6$ | $( 2, 3, 5)( 4, 7, 6)( 9,13,12,14,10,11)(16,20,19,21,17,18)$ |
| $ 3, 3, 3, 3, 3, 3, 3 $ | $196$ | $3$ | $( 1, 8,17)( 2, 9,21)( 3,10,18)( 4,11,15)( 5,12,19)( 6,13,16)( 7,14,20)$ |
| $ 21 $ | $588$ | $21$ | $( 1, 8,16, 6,13,15, 4,11,21, 2, 9,20, 7,14,19, 5,12,18, 3,10,17)$ |
| $ 21 $ | $588$ | $21$ | $( 1, 8,21, 2, 9,18, 3,10,15, 4,11,19, 5,12,16, 6,13,20, 7,14,17)$ |
| $ 21 $ | $588$ | $21$ | $( 1,17, 8, 6,20,11, 4,16,14, 2,19,10, 7,15,13, 5,18, 9, 3,21,12)$ |
| $ 21 $ | $588$ | $21$ | $( 1,16,14, 2,18, 9, 3,20,11, 4,15,13, 5,17, 8, 6,19,10, 7,21,12)$ |
| $ 3, 3, 3, 3, 3, 3, 3 $ | $196$ | $3$ | $( 1,21,12)( 2,16,14)( 3,18, 9)( 4,20,11)( 5,15,13)( 6,17, 8)( 7,19,10)$ |
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. |