# Properties

 Label 31T9 Degree $31$ Order $372000$ Cyclic no Abelian no Solvable no Primitive yes $p$-group no Group: $\PSL(3,5)$

## Group action invariants

 Degree $n$: $31$ Transitive number $t$: $9$ Group: $\PSL(3,5)$ Parity: $1$ Primitive: yes Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $1$ 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)

## Low degree resolvents

none

Resolvents shown for degrees $\leq 47$

## Subfields

Prime degree - none

## Low degree siblings

31T9

Siblings 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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1$ $775$ $2$ $( 1, 2)( 5,14)( 6,13)( 7,30)(10,12)(15,27)(16,31)(17,29)(19,23)(20,25)(21,24) (26,28)$ $3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1$ $15500$ $3$ $( 1,14,24)( 2, 5,21)( 3, 9, 8)( 6,23,25)( 7,16,27)(10,28,29)(11,18,22) (12,26,17)(13,19,20)(15,30,31)$ $4, 4, 4, 4, 4, 4, 1, 1, 1, 1, 1, 1, 1$ $775$ $4$ $( 1,27, 2,15)( 5,30,14, 7)( 6,29,13,17)(10,19,12,23)(16,21,31,24)(20,26,25,28)$ $4, 4, 4, 4, 4, 4, 1, 1, 1, 1, 1, 1, 1$ $775$ $4$ $( 1,15, 2,27)( 5, 7,14,30)( 6,17,13,29)(10,23,12,19)(16,24,31,21)(20,28,25,26)$ $6, 6, 6, 6, 3, 3, 1$ $15500$ $6$ $( 1,21,14, 2,24, 5)( 3, 8, 9)( 6,20,23,13,25,19)( 7,15,16,30,27,31) (10,17,28,12,29,26)(11,22,18)$ $12, 12, 3, 3, 1$ $15500$ $12$ $( 1,30,21,27,14,31, 2, 7,24,15, 5,16)( 3, 9, 8)( 6,12,20,29,23,26,13,10,25,17, 19,28)(11,18,22)$ $12, 12, 3, 3, 1$ $15500$ $12$ $( 1, 7,21,15,14,16, 2,30,24,27, 5,31)( 3, 9, 8)( 6,10,20,17,23,28,13,12,25,29, 19,26)(11,18,22)$ $8, 8, 8, 2, 2, 2, 1$ $15500$ $8$ $( 1,17,27, 6, 2,29,15,13)( 3,11)( 5,31,30,24,14,16, 7,21)( 8, 9) (10,26,19,25,12,28,23,20)(18,22)$ $8, 8, 8, 2, 2, 2, 1$ $15500$ $8$ $( 1,13,15,29, 2, 6,27,17)( 3,11)( 5,21, 7,16,14,24,30,31)( 8, 9) (10,20,23,28,12,25,19,26)(18,22)$ $24, 6, 1$ $15500$ $24$ $( 1,26, 5,17,19,31,27,25,30, 6,12,24, 2,28,14,29,23,16,15,20, 7,13,10,21) ( 3, 9,22,11, 8,18)$ $24, 6, 1$ $15500$ $24$ $( 1,25,14,13,19,24,15,26,30,29,10,31, 2,20, 5, 6,23,21,27,28, 7,17,12,16) ( 3, 9,22,11, 8,18)$ $24, 6, 1$ $15500$ $24$ $( 1,28, 5,29,19,16,27,20,30,13,12,21, 2,26,14,17,23,31,15,25, 7, 6,10,24) ( 3, 9,22,11, 8,18)$ $24, 6, 1$ $15500$ $24$ $( 1,20,14, 6,19,21,15,28,30,17,10,16, 2,25, 5,13,23,24,27,26, 7,29,12,31) ( 3, 9,22,11, 8,18)$ $31$ $12000$ $31$ $( 1,10,30,14,28, 4,15,25, 2,29,31,22, 7,24,26,27,18, 9,20,11,16,13,19,23,21, 12, 3, 6, 5,17, 8)$ $31$ $12000$ $31$ $( 1,18,10, 9,30,20,14,11,28,16, 4,13,15,19,25,23, 2,21,29,12,31, 3,22, 6, 7, 5,24,17,26, 8,27)$ $31$ $12000$ $31$ $( 1, 2,18,21,10,29, 9,12,30,31,20, 3,14,22,11, 6,28, 7,16, 5, 4,24,13,17,15, 26,19, 8,25,27,23)$ $31$ $12000$ $31$ $( 1,28, 2, 7,18,16,21, 5,10, 4,29,24, 9,13,12,17,30,15,31,26,20,19, 3, 8,14, 25,22,27,11,23, 6)$ $31$ $12000$ $31$ $( 1,30,28,15, 2,31, 7,26,18,20,16,19,21, 3, 5, 8,10,14, 4,25,29,22,24,27, 9, 11,13,23,12, 6,17)$ $31$ $12000$ $31$ $( 1, 8,17, 5, 6, 3,12,21,23,19,13,16,11,20, 9,18,27,26,24, 7,22,31,29, 2,25, 15, 4,28,14,30,10)$ $31$ $12000$ $31$ $( 1,27, 8,26,17,24, 5, 7, 6,22, 3,31,12,29,21, 2,23,25,19,15,13, 4,16,28,11, 14,20,30, 9,10,18)$ $31$ $12000$ $31$ $( 1,23,27,25, 8,19,26,15,17,13,24, 4, 5,16, 7,28, 6,11,22,14, 3,20,31,30,12, 9,29,10,21,18, 2)$ $31$ $12000$ $31$ $( 1, 6,23,11,27,22,25,14, 8, 3,19,20,26,31,15,30,17,12,13, 9,24,29, 4,10, 5, 21,16,18, 7, 2,28)$ $31$ $12000$ $31$ $( 1,17, 6,12,23,13,11, 9,27,24,22,29,25, 4,14,10, 8, 5, 3,21,19,16,20,18,26, 7,31, 2,15,28,30)$ $5, 5, 5, 5, 5, 1, 1, 1, 1, 1, 1$ $744$ $5$ $( 2, 6,24,18,26)( 3,15,20,30,21)( 4, 9,12,19,10)( 7,29,28,31,11) ( 8,13,14,27,16)$ $10, 10, 5, 2, 2, 1, 1$ $18600$ $10$ $( 1, 5)( 2,16, 6, 8,24,13,18,14,26,27)( 3, 9,15,12,20,19,30,10,21, 4) ( 7,31,29,11,28)(17,22)$ $5, 5, 5, 5, 5, 5, 1$ $14880$ $5$ $( 1, 5,17,25,22)( 2,29, 8,15, 9)( 3, 4,26, 7,16)( 6,28,13,20,12) (10,18,11,27,21)(14,30,19,24,31)$ $20, 5, 4, 1, 1$ $18600$ $20$ $( 1,17, 5,22)( 2,16,21,28,24,13,15,11,26,27,30,29, 6, 8, 3,31,18,14,20, 7) ( 4,19, 9,10,12)$ $20, 5, 4, 1, 1$ $18600$ $20$ $( 1,22, 5,17)( 2,29,21, 8,24,31,15,14,26, 7,30,16, 6,28, 3,13,18,11,20,27) ( 4,19, 9,10,12)$ $4, 4, 4, 4, 4, 4, 2, 2, 1, 1, 1$ $23250$ $4$ $( 1,27, 2,15)( 3,22)( 5, 7,14,30)( 6,16,13,31)( 8, 9)(10,28,12,26) (17,24,29,21)(19,20,23,25)$

## Group invariants

 Order: $372000=2^{5} \cdot 3 \cdot 5^{3} \cdot 31$ Cyclic: no Abelian: no Solvable: no GAP id: not available
 Character table: not available.