Group action invariants
| Degree $n$ : | $31$ | |
| Transitive number $t$ : | $9$ | |
| Group : | $\PSL(3,5)$ | |
| Parity: | $1$ | |
| Primitive: | Yes | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (4,11)(5,28,12,16)(6,22,13,8)(7,23,19,30)(9,17,18,29)(10,26,14,31)(15,27)(20,25,21,24), (1,14,22,5,4,31,19,23,7,24,6,9,28,27,26,20,17,30,21,8,12,11,3,2)(10,29,25,18,16,15) | |
| $|\Aut(F/K)|$: | $1$ |
Low degree resolvents
NoneResolvents shown for degrees $\leq 47$
Subfields
Prime degree - none
Low degree siblings
31T9Siblings are shown with degree $\leq 47$
A number field with this Galois group has exactly one arithmetically equivalent field.
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 $ | $1$ | $1$ | $()$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1 $ | $775$ | $2$ | $( 1,20)( 2,23)( 3, 4)( 5,19)( 8,31)( 9,16)(10,28)(11,21)(14,25)(17,29)(18,24) (27,30)$ |
| $ 4, 4, 4, 4, 4, 4, 2, 2, 1, 1, 1 $ | $23250$ | $4$ | $( 1,17,20,29)( 2,24,23,18)( 3,16, 4, 9)( 5,31,19, 8)( 7,13)(10,11,28,21) (12,22)(14,30,25,27)$ |
| $ 4, 4, 4, 4, 4, 4, 1, 1, 1, 1, 1, 1, 1 $ | $775$ | $4$ | $( 1,27, 4,11)( 3,21,20,30)( 5, 8,19,31)( 7,22,13,12)( 9,28,17,14)(10,29,25,16)$ |
| $ 4, 4, 4, 4, 4, 4, 1, 1, 1, 1, 1, 1, 1 $ | $775$ | $4$ | $( 1,11, 4,27)( 3,30,20,21)( 5,31,19, 8)( 7,12,13,22)( 9,14,17,28)(10,16,25,29)$ |
| $ 5, 5, 5, 5, 5, 1, 1, 1, 1, 1, 1 $ | $744$ | $5$ | $( 1,30,12,29, 8)( 2,23,26,18, 6)( 3, 7,25,19,27)( 4,21,22,16,31) ( 5,11,20,13,10)$ |
| $ 10, 10, 5, 2, 2, 1, 1 $ | $18600$ | $10$ | $( 1,16,30,31,12, 4,29,21, 8,22)( 2,18,23, 6,26)( 3, 5, 7,11,25,20,19,13,27,10) ( 9,17)(14,28)$ |
| $ 20, 5, 4, 1, 1 $ | $18600$ | $20$ | $( 1, 5,16, 7,30,11,31,25,12,20, 4,19,29,13,21,27, 8,10,22, 3)( 2, 6,18,26,23) ( 9,14,17,28)$ |
| $ 20, 5, 4, 1, 1 $ | $18600$ | $20$ | $( 1,19,16,13,30,27,31,10,12, 3, 4, 5,29, 7,21,11, 8,25,22,20)( 2, 6,18,26,23) ( 9,28,17,14)$ |
| $ 5, 5, 5, 5, 5, 5, 1 $ | $14880$ | $5$ | $( 1,29,30, 8,12)( 2,15,10,22, 3)( 4,25, 6, 9, 5)( 7,26,14,20,21) (11,16,27,18,28)(13,31,19,23,17)$ |
| $ 31 $ | $12000$ | $31$ | $( 1,12,19,28,16,17,15,25,22, 7,27, 6,20,31, 8, 4,21, 2,11,14,29,23,13, 3,30, 9,18, 5,26,24,10)$ |
| $ 31 $ | $12000$ | $31$ | $( 1,21,12, 2,19,11,28,14,16,29,17,23,15,13,25, 3,22,30, 7, 9,27,18, 6, 5,20, 26,31,24, 8,10, 4)$ |
| $ 31 $ | $12000$ | $31$ | $( 1,22,21,30,12, 7, 2, 9,19,27,11,18,28, 6,14, 5,16,20,29,26,17,31,23,24,15, 8,13,10,25, 4, 3)$ |
| $ 31 $ | $12000$ | $31$ | $( 1,16,22,20,21,29,30,26,12,17, 7,31, 2,23, 9,24,19,15,27, 8,11,13,18,10,28, 25, 6, 4,14, 3, 5)$ |
| $ 31 $ | $12000$ | $31$ | $( 1,19,16,15,22,27,20, 8,21,11,29,13,30,18,26,10,12,28,17,25, 7, 6,31, 4, 2, 14,23, 3, 9, 5,24)$ |
| $ 31 $ | $12000$ | $31$ | $( 1,10,24,26, 5,18, 9,30, 3,13,23,29,14,11, 2,21, 4, 8,31,20, 6,27, 7,22,25, 15,17,16,28,19,12)$ |
| $ 31 $ | $12000$ | $31$ | $( 1, 4,10, 8,24,31,26,20, 5, 6,18,27, 9, 7,30,22, 3,25,13,15,23,17,29,16,14, 28,11,19, 2,12,21)$ |
| $ 31 $ | $12000$ | $31$ | $( 1, 3, 4,25,10,13, 8,15,24,23,31,17,26,29,20,16, 5,14, 6,28,18,11,27,19, 9, 2, 7,12,30,21,22)$ |
| $ 31 $ | $12000$ | $31$ | $( 1, 5, 3,14, 4, 6,25,28,10,18,13,11, 8,27,15,19,24, 9,23, 2,31, 7,17,12,26, 30,29,21,20,22,16)$ |
| $ 31 $ | $12000$ | $31$ | $( 1,24, 5, 9, 3,23,14, 2, 4,31, 6, 7,25,17,28,12,10,26,18,30,13,29,11,21, 8, 20,27,22,15,16,19)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1 $ | $15500$ | $3$ | $( 1,19,16)( 2,22, 4)( 3,25,10)( 5,23,12)( 6,13,30)( 7,18,26)( 9,28,17) (11,20,21)(14,24,15)(27,29,31)$ |
| $ 6, 6, 6, 6, 3, 3, 1 $ | $15500$ | $6$ | $( 1,27,19,29,16,31)( 2,25,22,10, 4, 3)( 5,30,23, 6,12,13)( 7,20,18,21,26,11) ( 9,17,28)(14,15,24)$ |
| $ 8, 8, 8, 2, 2, 2, 1 $ | $15500$ | $8$ | $( 1,10,30, 7,29, 2,12,21)( 3, 6,18,31,22, 5,11,19)( 4,23,20,16,25,13,26,27) ( 9,15)(14,28)(17,24)$ |
| $ 8, 8, 8, 2, 2, 2, 1 $ | $15500$ | $8$ | $( 1,21,12, 2,29, 7,30,10)( 3,19,11, 5,22,31,18, 6)( 4,27,26,13,25,16,20,23) ( 9,15)(14,28)(17,24)$ |
| $ 12, 12, 3, 3, 1 $ | $15500$ | $12$ | $( 1, 5,27,30,19,23,29, 6,16,12,31,13)( 2,18,25,21,22,26,10,11, 4, 7, 3,20) ( 9,28,17)(14,24,15)$ |
| $ 12, 12, 3, 3, 1 $ | $15500$ | $12$ | $( 1, 6,27,12,19,13,29, 5,16,30,31,23)( 2,11,25, 7,22,20,10,18, 4,21, 3,26) ( 9,28,17)(14,24,15)$ |
| $ 24, 6, 1 $ | $15500$ | $24$ | $( 1,26, 5,10,27,11,30, 4,19, 7,23, 3,29,20, 6, 2,16,18,12,25,31,21,13,22) ( 9,24,28,15,17,14)$ |
| $ 24, 6, 1 $ | $15500$ | $24$ | $( 1, 4, 6,21,27, 3,12,26,19, 2,13,11,29,25, 5, 7,16,22,30,20,31,10,23,18) ( 9,24,28,15,17,14)$ |
| $ 24, 6, 1 $ | $15500$ | $24$ | $( 1,20, 5, 2,27,18,30,25,19,21,23,22,29,26, 6,10,16,11,12, 4,31, 7,13, 3) ( 9,24,28,15,17,14)$ |
| $ 24, 6, 1 $ | $15500$ | $24$ | $( 1,25, 6, 7,27,22,12,20,19,10,13,18,29, 4, 5,21,16, 3,30,26,31, 2,23,11) ( 9,24,28,15,17,14)$ |
Group invariants
| Order: | $372000=2^{5} \cdot 3 \cdot 5^{3} \cdot 31$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | No | |
| GAP id: | Data not available |
| Character table: Data not available. |