Group action invariants
| Degree $n$ : | $14$ | |
| Transitive number $t$ : | $42$ | |
| CHM label : | $2^{4}`L_{7}(14)$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,9,11)(2,4,8)(3,13,5)(6,12,10), (1,8)(2,9)(4,11), (1,8)(2,11)(3,10)(4,9)(5,6)(12,13), (1,3,5,7,9,11,13)(2,4,6,8,10,12,14) | |
| $|\Aut(F/K)|$: | $2$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ 168: $\GL(3,2)$ 336: 14T17 1344: 14T33 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 7: $\GL(3,2)$
Low degree siblings
14T42, 28T229 x 2, 28T230, 28T231 x 2, 42T327 x 2, 42T330 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 $ | $1$ | $1$ | $()$ |
| $ 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $7$ | $2$ | $( 4,11)( 5,12)( 7,14)$ |
| $ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $7$ | $2$ | $( 2, 9)( 4,11)( 5,12)( 6,13)$ |
| $ 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 8)( 2, 9)( 3,10)( 4,11)( 5,12)( 6,13)( 7,14)$ |
| $ 4, 4, 2, 2, 2 $ | $42$ | $4$ | $( 1, 8)( 2, 9)( 3, 5,10,12)( 4,11)( 6, 7,13,14)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1 $ | $84$ | $2$ | $( 1, 8)( 2, 9)( 3,12)( 5,10)( 6,14)( 7,13)$ |
| $ 4, 4, 2, 2, 1, 1 $ | $42$ | $4$ | $( 1, 8)( 3, 5,10,12)( 4,11)( 6,14,13, 7)$ |
| $ 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $84$ | $2$ | $( 1, 8)( 3,12)( 5,10)( 6, 7)(13,14)$ |
| $ 4, 4, 2, 1, 1, 1, 1 $ | $42$ | $4$ | $( 2, 9)( 3,12,10, 5)( 6, 7,13,14)$ |
| $ 4, 4, 1, 1, 1, 1, 1, 1 $ | $42$ | $4$ | $( 3,12,10, 5)( 6,14,13, 7)$ |
| $ 8, 4, 2 $ | $168$ | $8$ | $( 1, 8)( 2,10, 4,14, 9, 3,11, 7)( 5, 6,12,13)$ |
| $ 8, 2, 2, 2 $ | $168$ | $8$ | $( 1, 8)( 2,10,11,14, 9, 3, 4, 7)( 5, 6)(12,13)$ |
| $ 8, 4, 1, 1 $ | $168$ | $8$ | $( 2, 3,11,14, 9,10, 4, 7)( 5, 6,12,13)$ |
| $ 8, 2, 2, 1, 1 $ | $168$ | $8$ | $( 2, 3, 4,14, 9,10,11, 7)( 5, 6)(12,13)$ |
| $ 3, 3, 3, 3, 1, 1 $ | $224$ | $3$ | $( 2, 3,12)( 4, 7, 6)( 5, 9,10)(11,14,13)$ |
| $ 6, 3, 3, 1, 1 $ | $224$ | $6$ | $( 2, 3, 5, 9,10,12)( 4,14,13)( 6,11, 7)$ |
| $ 6, 6, 2 $ | $224$ | $6$ | $( 1, 8)( 2,10, 5, 9, 3,12)( 4,14,13,11, 7, 6)$ |
| $ 6, 3, 3, 2 $ | $224$ | $6$ | $( 1, 8)( 2,10,12)( 3, 5, 9)( 4, 7, 6,11,14,13)$ |
| $ 7, 7 $ | $192$ | $7$ | $( 1, 2, 3, 4,12, 6, 7)( 5,13,14, 8, 9,10,11)$ |
| $ 14 $ | $192$ | $14$ | $( 1, 2, 3,11,12, 6,14, 8, 9,10, 4, 5,13, 7)$ |
| $ 14 $ | $192$ | $14$ | $( 1, 9, 3,14,13,11,12, 8, 2,10, 7, 6, 4, 5)$ |
| $ 7, 7 $ | $192$ | $7$ | $( 1, 9, 3, 7, 6,11, 5)( 2,10,14,13, 4,12, 8)$ |
Group invariants
| Order: | $2688=2^{7} \cdot 3 \cdot 7$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | No | |
| GAP id: | Data not available |
| Character table: Data not available. |