Group action invariants
| Degree $n$: | $31$ | |
| Transitive number $t$: | $10$ | |
| Group: | $\PSL(5,2)$ | |
| Parity: | $1$ | |
| Primitive: | yes | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| $|\Aut(F/K)|$: | $1$ | |
| Generators: | (1,19)(6,12)(9,24)(11,18)(15,16)(17,26)(23,27)(28,30), (1,5,4,3,2)(6,31,30,8,7)(9,29,28,23,10)(11,22,21,20,19)(12,18,26,14,13)(15,25,24,27,17) |
Low degree resolvents
noneResolvents shown for degrees $\leq 47$
Subfields
Prime degree - none
Low degree siblings
31T10Siblings 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $465$ | $2$ | $( 2,31)( 4,22)( 9,15)(12,27)(14,21)(18,19)(20,29)(26,30)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1 $ | $19840$ | $3$ | $( 1,17,16)( 2,14,20)( 3,25,10)( 7, 8,13)(11,28,24)(12,26,19)(18,27,30) (21,29,31)$ |
| $ 6, 6, 3, 3, 3, 3, 2, 2, 1, 1, 1 $ | $416640$ | $6$ | $( 1,16,17)( 2,29,14,31,20,21)( 3,10,25)( 4,22)( 7,13, 8)( 9,15)(11,24,28) (12,18,26,27,19,30)$ |
| $ 7, 7, 7, 7, 1, 1, 1 $ | $238080$ | $7$ | $( 1,28,13, 7, 6, 3,24)( 2,19, 4,20,21,30, 9)( 8,16,11,17,25,10,23) (14,26,15,31,18,22,29)$ |
| $ 7, 7, 7, 7, 1, 1, 1 $ | $238080$ | $7$ | $( 1,24, 3, 6, 7,13,28)( 2, 9,30,21,20, 4,19)( 8,23,10,25,17,11,16) (14,29,22,18,31,15,26)$ |
| $ 14, 7, 7, 2, 1 $ | $714240$ | $14$ | $( 1, 6,28, 3,13,24, 7)( 2,14,19,26, 4,15,20,31,21,18,30,22, 9,29) ( 8,25,16,10,11,23,17)(12,27)$ |
| $ 14, 7, 7, 2, 1 $ | $714240$ | $14$ | $( 1, 3, 7,28,24, 6,13)( 2,26,20,18, 9,14, 4,31,30,29,19,15,21,22) ( 8,10,17,16,23,25,11)(12,27)$ |
| $ 4, 4, 4, 4, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1 $ | $26040$ | $4$ | $( 1,15,20,10)( 2,26, 6,25)( 3,19)( 7,11, 9,29)( 8,18)(13,31,30,23)(14,17) (21,28)$ |
| $ 12, 6, 4, 3, 3, 2, 1 $ | $833280$ | $12$ | $( 1,10,20,15)( 2,11,31,26, 9,30, 6,29,23,25, 7,13)( 3,21,18,19,28, 8) ( 4,16, 5)(14,17)(22,27,24)$ |
| $ 5, 5, 5, 5, 5, 5, 1 $ | $666624$ | $5$ | $( 1,19, 8,16,20)( 2,27,28,26,12)( 3,29, 4, 9,22)( 5,30,13,21,17) ( 6,23,14,31,25)( 7,10,18,15,11)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1 $ | $55552$ | $3$ | $( 1, 7, 3)( 2, 5,31)( 4, 8,18)( 6,28,13)( 9,16,15)(10,29,19)(11,22,20) (12,17,14)(21,23,26)(25,27,30)$ |
| $ 15, 15, 1 $ | $666624$ | $15$ | $( 1,29,18,16,22, 7,19, 4,15,20, 3,10, 8, 9,11)( 2,25,13,26,14, 5,27, 6,21,12, 31,30,28,23,17)$ |
| $ 15, 15, 1 $ | $666624$ | $15$ | $( 1,10, 4,16,11, 3,19,18, 9,20, 7,29, 8,15,22)( 2,30, 6,26,17,31,27,13,23,12, 5,25,28,21,14)$ |
| $ 31 $ | $322560$ | $31$ | $( 1, 3, 8,10,31,28,22,27,17,29,16, 2, 6, 5,13, 4,26,18,24,21, 9,30,15,12,23, 25,20,11, 7,14,19)$ |
| $ 31 $ | $322560$ | $31$ | $( 1,25,21, 5,27, 3,20, 9,13,17, 8,11,30, 4,29,10, 7,15,26,16,31,14,12,18, 2, 28,19,23,24, 6,22)$ |
| $ 31 $ | $322560$ | $31$ | $( 1,28,16, 4, 9,25,19,31,29,13,21,23,14,10,17, 5,24,12, 7, 8,27, 6,18,15,11, 3,22, 2,26,30,20)$ |
| $ 31 $ | $322560$ | $31$ | $( 1,19,14, 7,11,20,25,23,12,15,30, 9,21,24,18,26, 4,13, 5, 6, 2,16,29,17,27, 22,28,31,10, 8, 3)$ |
| $ 31 $ | $322560$ | $31$ | $( 1,22, 6,24,23,19,28, 2,18,12,14,31,16,26,15, 7,10,29, 4,30,11, 8,17,13, 9, 20, 3,27, 5,21,25)$ |
| $ 31 $ | $322560$ | $31$ | $( 1,20,30,26, 2,22, 3,11,15,18, 6,27, 8, 7,12,24, 5,17,10,14,23,21,13,29,31, 19,25, 9, 4,16,28)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1 $ | $6510$ | $2$ | $( 2,21)( 3,23)( 4,27)( 6, 8)( 7,17)( 9,19)(10,28)(12,22)(14,31)(15,18)(20,26) (29,30)$ |
| $ 6, 6, 6, 6, 3, 3, 1 $ | $833280$ | $6$ | $( 1, 5,25)( 2,12,23,21,22, 3)( 4,10,19,27,28, 9)( 6,31,26, 8,14,20) ( 7,15,29,17,18,30)(11,13,16)$ |
| $ 21, 7, 3 $ | $476160$ | $21$ | $( 1, 7,23, 5, 9,28,16, 6,15,20, 8,10,22,17,31,13, 2,19,30,14,21)( 3,27,18) ( 4,29,26,11,12,25,24)$ |
| $ 21, 7, 3 $ | $476160$ | $21$ | $( 1, 8,21,20,14,15,30, 6,19,16, 2,28,13, 9,31, 5,17,23,22, 7,10)( 3,27,18) ( 4,11,24,26,25,29,12)$ |
| $ 4, 4, 4, 4, 2, 2, 2, 2, 2, 2, 1, 1, 1 $ | $78120$ | $4$ | $( 1, 7,22,18)( 2,16,17,20)( 3,15)( 4,12)( 5, 8,30, 9)( 6,27,14,13)(10,21) (11,25)(19,23)(28,31)$ |
| $ 8, 8, 4, 4, 4, 2, 1 $ | $624960$ | $8$ | $( 1, 2, 7,16,22,17,18,20)( 3,28,15,31)( 4,23,12,19)( 5,13, 8, 6,30,27, 9,14) (10,11,21,25)(24,29)$ |
| $ 4, 4, 4, 4, 4, 4, 2, 2, 1, 1, 1 $ | $312480$ | $4$ | $( 1,25)( 2, 6,18,28)( 3, 9, 7,14)( 4,26,27,20)( 8,15,10,21)(11,13) (12,29,22,30)(17,31,23,19)$ |
Group invariants
| Order: | $9999360=2^{10} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 31$ | |
| Cyclic: | no | |
| Abelian: | no | |
| Solvable: | no | |
| GAP id: | not available |
| Character table: not available. |