Group action invariants
| Degree $n$ : | $35$ | |
| Transitive number $t$ : | $26$ | |
| Parity: | $1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,12,23,31,6,20,26,4,14,21,35,10,17,29,2,13,22,33,8,19,28,5,11,25,34,7,18,27,3,15,24,32,9,16,30), (1,28,13,7,20,35)(2,27,14,10,19,33,3,26,15,9,16,32)(4,29,12,6,17,31,5,30,11,8,18,34)(21,23,25,22) | |
| $|\Aut(F/K)|$: | $1$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ 3: $C_3$ 6: $C_6$ 42: $F_7$ 120: $S_5$ 360: $S_5 \times C_3$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 5: $S_5$
Degree 7: $F_7$
Low degree siblings
42T298Siblings 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ |
| $ 7, 7, 7, 7, 7 $ | $6$ | $7$ | $( 1, 7,13,20,24,28,35)( 2, 6,15,19,21,30,34)( 3, 8,14,16,25,29,31) ( 4, 9,11,17,23,27,33)( 5,10,12,18,22,26,32)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1 $ | $7$ | $3$ | $( 6,21,15)( 7,24,13)( 8,25,14)( 9,23,11)(10,22,12)(16,29,31)(17,27,33) (18,26,32)(19,30,34)(20,28,35)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1 $ | $7$ | $3$ | $( 6,15,21)( 7,13,24)( 8,14,25)( 9,11,23)(10,12,22)(16,31,29)(17,33,27) (18,32,26)(19,34,30)(20,35,28)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1 $ | $70$ | $2$ | $( 1,22)( 2,21)( 3,25)( 4,23)( 5,24)( 6,19)( 7,18)( 8,16)( 9,17)(10,20)(12,13) (26,35)(27,33)(28,32)(29,31)(30,34)$ |
| $ 6, 6, 6, 6, 6, 2, 1, 1, 1 $ | $70$ | $6$ | $( 1,12, 7,26,20,22)( 2,15, 6,30,19,21)( 3,14, 8,29,16,25)( 4,11, 9,27,17,23) ( 5,13,10,28,18,24)(32,35)$ |
| $ 6, 6, 6, 6, 6, 2, 1, 1, 1 $ | $70$ | $6$ | $( 1,10,35,18,13,22)( 2, 6,34,19,15,21)( 3, 8,31,16,14,25)( 4, 9,33,17,11,23) ( 5, 7,32,20,12,24)(26,28)$ |
| $ 14, 14, 7 $ | $90$ | $14$ | $( 1,26,20,10,35,22,13, 5,28,18, 7,32,24,12)( 2,27,19, 9,34,23,15, 4,30,17, 6, 33,21,11)( 3,29,16, 8,31,25,14)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1 $ | $15$ | $2$ | $( 1, 5)( 2, 4)( 6, 9)( 7,10)(11,15)(12,13)(17,19)(18,20)(21,23)(22,24)(26,28) (27,30)(32,35)(33,34)$ |
| $ 6, 6, 6, 6, 3, 3, 2, 2, 1 $ | $105$ | $6$ | $( 1,32,13, 5,35,12)( 2,33,15, 4,34,11)( 3,31,14)( 6,17,21, 9,19,23) ( 7,18,24,10,20,22)( 8,16,25)(26,28)(27,30)$ |
| $ 6, 6, 6, 6, 3, 3, 2, 2, 1 $ | $105$ | $6$ | $( 1,18,13, 5,20,12)( 2,17,15, 4,19,11)( 3,16,14)( 6,27,34, 9,30,33) ( 7,26,35,10,28,32)( 8,29,31)(21,23)(22,24)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ | $140$ | $3$ | $( 1,26, 9)( 2,30, 6)( 3,29, 8)( 4,28,10)( 5,27, 7)(11,13,12)(16,25,31) (17,24,32)(18,23,35)(19,21,34)(20,22,33)$ |
| $ 21, 7, 7 $ | $120$ | $21$ | $( 1,32,27,24,18,11, 7, 5,33,28,22,17,13,10, 4,35,26,23,20,12, 9) ( 2,34,30,21,19,15, 6)( 3,31,29,25,16,14, 8)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $20$ | $3$ | $( 1, 5, 4)( 7,10, 9)(11,13,12)(17,20,18)(22,23,24)(26,27,28)(32,33,35)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ | $140$ | $3$ | $( 1,18, 9)( 2,19, 6)( 3,16, 8)( 4,20,10)( 5,17, 7)(11,24,26)(12,23,28) (13,22,27)(14,25,29)(15,21,30)(32,33,35)$ |
| $ 6, 6, 6, 3, 2, 2, 2, 2, 2, 2, 2 $ | $140$ | $6$ | $( 1,26, 4,28, 5,27)( 2,29)( 3,30)( 6,25)( 7,22, 9,24,10,23)( 8,21) (11,20,12,17,13,18)(14,19)(15,16)(31,34)(32,33,35)$ |
| $ 6, 6, 6, 6, 6, 3, 2 $ | $140$ | $6$ | $( 1,32,17, 7,12,27)( 2,31,19, 8,15,29)( 3,34,16, 6,14,30)( 4,35,18, 9,13,26) ( 5,33,20,10,11,28)(21,25)(22,23,24)$ |
| $ 6, 6, 6, 6, 6, 3, 2 $ | $140$ | $6$ | $( 1,18,23,13,32,27)( 2,16,21,14,34,29)( 3,19,25,15,31,30)( 4,20,22,11,35,26) ( 5,17,24,12,33,28)( 6, 8)( 7,10, 9)$ |
| $ 12, 12, 6, 4, 1 $ | $210$ | $12$ | $( 1,18,23,15,35,26, 4,19,24,12,33,30)( 2,20,22,11,34,28, 5,17,21,13,32,27) ( 3,16,25,14,31,29)( 6, 7,10, 9)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 2, 2, 2, 1 $ | $210$ | $4$ | $( 1,26, 4,30)( 2,28, 5,27)( 3,29)( 6,24,10,23)( 7,22, 9,21)( 8,25) (11,19,13,18)(12,17,15,20)(14,16)(32,33,34,35)$ |
| $ 12, 12, 6, 4, 1 $ | $210$ | $12$ | $( 1,32,17, 6,13,26, 4,34,20,10,11,30)( 2,35,18, 9,15,28, 5,33,19, 7,12,27) ( 3,31,16, 8,14,29)(21,24,22,23)$ |
| $ 15, 15, 5 $ | $168$ | $15$ | $( 1,18, 9, 2,16, 7, 5,17, 6, 3,20,10, 4,19, 8)(11,21,29,13,22,27,15,25,28,12, 23,30,14,24,26)(31,35,32,33,34)$ |
| $ 15, 15, 5 $ | $168$ | $15$ | $( 1,26, 9, 2,29, 7, 5,27, 6, 3,28,10, 4,30, 8)(11,15,14,13,12)(16,24,32,17,21, 31,20,22,33,19,25,35,18,23,34)$ |
| $ 35 $ | $72$ | $35$ | $( 1,32,27,21,16,13,10, 4,34,29,24,18,11, 6, 3,35,26,23,19,14, 7, 5,33,30,25, 20,12, 9, 2,31,28,22,17,15, 8)$ |
| $ 5, 5, 5, 5, 5, 5, 5 $ | $24$ | $5$ | $( 1, 5, 4, 2, 3)( 6, 8, 7,10, 9)(11,15,14,13,12)(16,20,18,17,19) (21,25,24,22,23)(26,27,30,29,28)(31,35,32,33,34)$ |
| $ 35 $ | $72$ | $35$ | $( 1,12,23,34, 8,20,26, 4,15,25,35,10,17,30, 3,13,22,33, 6,16,28, 5,11,21,31, 7,18,27, 2,14,24,32, 9,19,29)$ |
Group invariants
| Order: | $2520=2^{3} \cdot 3^{2} \cdot 5 \cdot 7$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | No | |
| GAP id: | Data not available |
| Character table: Data not available. |