Group action invariants
| Degree $n$ : | $35$ | |
| Transitive number $t$ : | $25$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,10,13,18,24,26,35,5,7,12,20,22,28,32)(2,9,14,19,23,29,34,4,8,15,17,25,30,33,3,6,11,16,21,27,31), (1,25,26,4,24,29,5,23,28,3,22,27)(2,21,30)(6,34,15)(7,31,12,9,35,14,10,33,13,8,32,11)(16,18,17,20) | |
| $|\Aut(F/K)|$: | $1$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ 3: $C_3$ 6: $C_6$ 21: $C_7:C_3$ 42: $(C_7:C_3) \times C_2$ 120: $S_5$ 360: $S_5 \times C_3$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 5: $S_5$
Degree 7: $C_7:C_3$
Low degree siblings
42T296Siblings 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 $ | $3$ | $7$ | $( 1,13,24,35, 7,20,28)( 2,15,21,34, 6,19,30)( 3,14,25,31, 8,16,29) ( 4,11,23,33, 9,17,27)( 5,12,22,32,10,18,26)$ |
| $ 7, 7, 7, 7, 7 $ | $3$ | $7$ | $( 1,35,28,24,20,13, 7)( 2,34,30,21,19,15, 6)( 3,31,29,25,16,14, 8) ( 4,33,27,23,17,11, 9)( 5,32,26,22,18,12,10)$ |
| $ 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 $ | $7$ | $3$ | $( 6,15,21)( 7,13,24)( 8,14,25)( 9,11,23)(10,12,22)(16,31,29)(17,33,27) (18,32,26)(19,34,30)(20,35,28)$ |
| $ 6, 6, 3, 3, 3, 3, 3, 3, 2, 1, 1, 1 $ | $70$ | $6$ | $( 1,32,24, 5,35,22)( 2,34,21)( 3,31,25)( 4,33,23)( 7,10)(11,17,27) (12,20,26,13,18,28)(14,16,29)(15,19,30)$ |
| $ 14, 7, 7, 7 $ | $30$ | $14$ | $( 1,18,35,12,28,10,24, 5,20,32,13,26, 7,22)( 2,19,34,15,30, 6,21) ( 3,16,31,14,29, 8,25)( 4,17,33,11,27, 9,23)$ |
| $ 14, 7, 7, 7 $ | $30$ | $14$ | $( 1,12,24,32, 7,18,28, 5,13,22,35,10,20,26)( 2,15,21,34, 6,19,30) ( 3,14,25,31, 8,16,29)( 4,11,23,33, 9,17,27)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $10$ | $2$ | $( 1, 5)( 7,10)(12,13)(18,20)(22,24)(26,28)(32,35)$ |
| $ 6, 6, 3, 3, 3, 3, 3, 3, 2, 1, 1, 1 $ | $70$ | $6$ | $( 1,26,24, 5,28,22)( 2,30,21)( 3,29,25)( 4,27,23)( 6,15,34)( 7,12,35,10,13,32) ( 8,14,31)( 9,11,33)(18,20)$ |
| $ 14, 14, 7 $ | $45$ | $14$ | $( 1,32,28,22,20,12, 7, 5,35,26,24,18,13,10)( 2,34,30,21,19,15, 6) ( 3,33,29,23,16,11, 8, 4,31,27,25,17,14, 9)$ |
| $ 14, 14, 7 $ | $45$ | $14$ | $( 1,10,13,18,24,26,35, 5, 7,12,20,22,28,32)( 2, 6,15,19,21,30,34) ( 3, 9,14,17,25,27,31, 4, 8,11,16,23,29,33)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1 $ | $15$ | $2$ | $( 1, 5)( 3, 4)( 7,10)( 8, 9)(11,14)(12,13)(16,17)(18,20)(22,24)(23,25)(26,28) (27,29)(31,33)(32,35)$ |
| $ 6, 6, 6, 6, 3, 3, 2, 2, 1 $ | $105$ | $6$ | $( 1,18, 7, 5,20,10)( 2,19, 6)( 3,17, 8, 4,16, 9)(11,25,27,14,23,29) (12,24,26,13,22,28)(15,21,30)(31,33)(32,35)$ |
| $ 6, 6, 6, 6, 3, 3, 2, 2, 1 $ | $105$ | $6$ | $( 1,26, 7, 5,28,10)( 2,30, 6)( 3,27, 8, 4,29, 9)(11,14)(12,13)(16,23,31,17,25, 33)(18,24,32,20,22,35)(19,21,34)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ | $140$ | $3$ | $( 1,12,31)( 2,15,34)( 3,13,32)( 4,11,33)( 5,14,35)( 6,21,19)( 7,22,16) ( 8,24,18)( 9,23,17)(10,25,20)(26,29,28)$ |
| $ 21, 7, 7 $ | $60$ | $21$ | $( 1,10,14,20,22,29,35, 5, 8,13,18,25,28,32, 3, 7,12,16,24,26,31) ( 2, 6,15,19,21,30,34)( 4, 9,11,17,23,27,33)$ |
| $ 21, 7, 7 $ | $60$ | $21$ | $( 1,18,31,13,26, 8,24, 5,16,35,12,29, 7,22, 3,20,32,14,28,10,25) ( 2,19,34,15,30, 6,21)( 4,17,33,11,27, 9,23)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $20$ | $3$ | $( 1, 5, 3)( 7,10, 8)(12,14,13)(16,20,18)(22,25,24)(26,29,28)(31,35,32)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ | $140$ | $3$ | $( 1,22,31)( 2,21,34)( 3,24,32)( 4,23,33)( 5,25,35)( 7,10, 8)(11,27,17) (12,29,20)(13,26,16)(14,28,18)(15,30,19)$ |
| $ 6, 6, 3, 3, 3, 3, 3, 3, 3, 2 $ | $140$ | $6$ | $( 1,12,31)( 2,11,34, 4,15,33)( 3,13,32)( 5,14,35)( 6,23,19, 9,21,17)( 7,22,16) ( 8,24,18)(10,25,20)(26,29,28)(27,30)$ |
| $ 21, 14 $ | $60$ | $42$ | $( 1,10,14,20,22,29,35, 5, 8,13,18,25,28,32, 3, 7,12,16,24,26,31) ( 2, 9,15,17,21,27,34, 4, 6,11,19,23,30,33)$ |
| $ 21, 14 $ | $60$ | $42$ | $( 1,18,31,13,26, 8,24, 5,16,35,12,29, 7,22, 3,20,32,14,28,10,25) ( 2,17,34,11,30, 9,21, 4,19,33,15,27, 6,23)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2 $ | $20$ | $6$ | $( 1, 5, 3)( 2, 4)( 6, 9)( 7,10, 8)(11,15)(12,14,13)(16,20,18)(17,19)(21,23) (22,25,24)(26,29,28)(27,30)(31,35,32)(33,34)$ |
| $ 6, 6, 3, 3, 3, 3, 3, 3, 3, 2 $ | $140$ | $6$ | $( 1,22,31)( 2,23,34, 4,21,33)( 3,24,32)( 5,25,35)( 6, 9)( 7,10, 8) (11,30,17,15,27,19)(12,29,20)(13,26,16)(14,28,18)$ |
| $ 12, 12, 4, 3, 3, 1 $ | $210$ | $12$ | $( 1,12,31, 4,13,32, 3,11,35, 5,14,33)( 2,15,34)( 6,21,19)( 7,22,16, 9,24,18, 8,23,20,10,25,17)(26,29,27,28)$ |
| $ 28, 7 $ | $90$ | $28$ | $( 1,10,14,17,24,26,31, 4, 7,12,16,23,28,32, 3, 9,13,18,25,27,35, 5, 8,11,20, 22,29,33)( 2, 6,15,19,21,30,34)$ |
| $ 28, 7 $ | $90$ | $28$ | $( 1,18,31,11,28,10,25, 4,20,32,14,27, 7,22, 3,17,35,12,29, 9,24, 5,16,33,13, 26, 8,23)( 2,19,34,15,30, 6,21)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 1, 1, 1, 1, 1, 1, 1 $ | $30$ | $4$ | $( 1, 5, 3, 4)( 7,10, 8, 9)(11,13,12,14)(16,17,20,18)(22,25,23,24)(26,29,27,28) (31,33,35,32)$ |
| $ 12, 12, 4, 3, 3, 1 $ | $210$ | $12$ | $( 1,22,31, 4,24,32, 3,23,35, 5,25,33)( 2,21,34)( 7,10, 8, 9)(11,28,18,14,27, 20,12,29,17,13,26,16)(15,30,19)$ |
| $ 15, 15, 5 $ | $168$ | $15$ | $( 1,12,31, 4,15,35, 5,14,33, 2,13,32, 3,11,34)( 6,24,18, 8,23,19, 7,22,16, 9, 21,20,10,25,17)(26,29,27,30,28)$ |
| $ 35 $ | $72$ | $35$ | $( 1,10,14,17,21,28,32, 3, 9,15,20,22,29,33, 2, 7,12,16,23,30,35, 5, 8,11,19, 24,26,31, 4, 6,13,18,25,27,34)$ |
| $ 35 $ | $72$ | $35$ | $( 1,18,31,11,30, 7,22, 3,17,34,13,26, 8,23, 2,20,32,14,27, 6,24, 5,16,33,15, 28,10,25, 4,19,35,12,29, 9,21)$ |
| $ 5, 5, 5, 5, 5, 5, 5 $ | $24$ | $5$ | $( 1, 5, 3, 4, 2)( 6, 7,10, 8, 9)(11,15,13,12,14)(16,17,19,20,18) (21,24,22,25,23)(26,29,27,30,28)(31,33,34,35,32)$ |
| $ 15, 15, 5 $ | $168$ | $15$ | $( 1,22,31, 4,21,35, 5,25,33, 2,24,32, 3,23,34)( 6, 7,10, 8, 9)(11,30,20,12,29, 17,15,28,18,14,27,19,13,26,16)$ |
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. |