Group action invariants
| Degree $n$ : | $14$ | |
| Transitive number $t$ : | $36$ | |
| CHM label : | $1/2[F_{42}(7)^{2}]2$ | |
| Parity: | $1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (2,4,6,8,10,12,14), (2,4,8)(6,12,10), (1,6,13,8)(2,9,12,5)(3,4,11,10)(7,14), (1,13)(2,12)(3,11)(4,10)(5,9)(6,8) | |
| $|\Aut(F/K)|$: | $1$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ 3: $C_3$ 4: $C_4$ 6: $S_3$, $C_6$ 12: $C_{12}$, $C_3 : C_4$ 18: $S_3\times C_3$ 36: $C_3\times (C_3 : C_4)$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 7: None
Low degree siblings
28T169, 42T248, 42T249, 42T250, 42T251, 42T257Siblings 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 $ | $1$ | $1$ | $()$ |
| $ 7, 1, 1, 1, 1, 1, 1, 1 $ | $12$ | $7$ | $( 2, 4, 6, 8,10,12,14)$ |
| $ 7, 7 $ | $36$ | $7$ | $( 1, 3, 5, 7, 9,11,13)( 2, 4, 6, 8,10,12,14)$ |
| $ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1 $ | $14$ | $3$ | $( 4, 6,10)( 8,14,12)$ |
| $ 7, 3, 3, 1 $ | $84$ | $21$ | $( 1, 3, 5, 7, 9,11,13)( 4, 6,10)( 8,14,12)$ |
| $ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1 $ | $14$ | $3$ | $( 4,10, 6)( 8,12,14)$ |
| $ 7, 3, 3, 1 $ | $84$ | $21$ | $( 1, 3, 5, 7, 9,11,13)( 4,10, 6)( 8,12,14)$ |
| $ 3, 3, 3, 3, 1, 1 $ | $49$ | $3$ | $( 3, 5, 9)( 4, 6,10)( 7,13,11)( 8,14,12)$ |
| $ 3, 3, 3, 3, 1, 1 $ | $98$ | $3$ | $( 3, 5, 9)( 4,10, 6)( 7,13,11)( 8,12,14)$ |
| $ 3, 3, 3, 3, 1, 1 $ | $49$ | $3$ | $( 3, 9, 5)( 4,10, 6)( 7,11,13)( 8,12,14)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1 $ | $49$ | $2$ | $( 3,13)( 4,14)( 5,11)( 6,12)( 7, 9)( 8,10)$ |
| $ 6, 2, 2, 2, 1, 1 $ | $98$ | $6$ | $( 3,13)( 4,12,10,14, 6, 8)( 5,11)( 7, 9)$ |
| $ 6, 2, 2, 2, 1, 1 $ | $98$ | $6$ | $( 3,13)( 4, 8, 6,14,10,12)( 5,11)( 7, 9)$ |
| $ 6, 6, 1, 1 $ | $49$ | $6$ | $( 3,11, 9,13, 5, 7)( 4,12,10,14, 6, 8)$ |
| $ 6, 6, 1, 1 $ | $98$ | $6$ | $( 3,11, 9,13, 5, 7)( 4, 8, 6,14,10,12)$ |
| $ 6, 6, 1, 1 $ | $49$ | $6$ | $( 3, 7, 5,13, 9,11)( 4, 8, 6,14,10,12)$ |
| $ 4, 4, 4, 2 $ | $147$ | $4$ | $( 1, 6,13, 8)( 2, 9,12, 5)( 3, 4,11,10)( 7,14)$ |
| $ 12, 2 $ | $147$ | $12$ | $( 1,10, 3, 6,13,14, 7,12, 5, 2, 9, 8)( 4,11)$ |
| $ 12, 2 $ | $147$ | $12$ | $( 1, 4,11, 6,13,12, 5, 2, 9,14, 7, 8)( 3,10)$ |
| $ 4, 4, 4, 2 $ | $147$ | $4$ | $( 1,12,11, 8)( 2, 7, 4, 5)( 3,14, 9, 6)(10,13)$ |
| $ 12, 2 $ | $147$ | $12$ | $( 1, 8)( 2, 7, 6, 3,12,11,14, 9,10,13, 4, 5)$ |
| $ 12, 2 $ | $147$ | $12$ | $( 1,14, 9, 4, 5, 2, 7,10,13, 6, 3, 8)(11,12)$ |
Group invariants
| Order: | $1764=2^{2} \cdot 3^{2} \cdot 7^{2}$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | [1764, 133] |
| Character table: Data not available. |