# Properties

 Label 31T10 Order $$9999360$$ n $$31$$ Cyclic No Abelian No Solvable No Primitive Yes $p$-group No Group: $\PSL(5,2)$

## Group action invariants

 Degree $n$ : $31$ Transitive number $t$ : $10$ Group : $\PSL(5,2)$ Parity: $1$ Primitive: Yes Nilpotency class: $-1$ (not nilpotent) 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) $|\Aut(F/K)|$: $1$

## Low degree resolvents

None

Resolvents shown for degrees $\leq 47$

## Subfields

Prime degree - none

## Low degree siblings

31T10

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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $31$ $322560$ $31$ $( 1, 3,12,22,30,20,14,13,25,23, 4, 2,17,29,27,18, 8,21, 6,19,16,11,24,10,31, 5,28,15, 7,26, 9)$ $31$ $322560$ $31$ $( 1, 5,19,29,13, 3,28,16,27,25,12,15,11,18,23,22, 7,24, 8, 4,30,26,10,21, 2, 20, 9,31, 6,17,14)$ $31$ $322560$ $31$ $( 1,20, 4,18,16, 5, 9,30,23,27,19,31,26,22,25,29, 6,10, 7,12,13,17,21,24,15, 3,14, 2, 8,11,28)$ $31$ $322560$ $31$ $( 1, 9,26, 7,15,28, 5,31,10,24,11,16,19, 6,21, 8,18,27,29,17, 2, 4,23,25,13, 14,20,30,22,12, 3)$ $31$ $322560$ $31$ $( 1,14,17, 6,31, 9,20, 2,21,10,26,30, 4, 8,24, 7,22,23,18,11,15,12,25,27,16, 28, 3,13,29,19, 5)$ $31$ $322560$ $31$ $( 1,28,11, 8, 2,14, 3,15,24,21,17,13,12, 7,10, 6,29,25,22,26,31,19,27,23,30, 9, 5,16,18, 4,20)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1$ $6510$ $2$ $( 1,21)( 3,20)( 4,12)( 6, 7)( 8,29)(10,23)(11,14)(13,19)(17,26)(18,25)(22,27) (28,30)$ $3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1$ $55552$ $3$ $( 1, 6,29)( 2, 9,24)( 3,18,27)( 4,26,19)( 5,16,15)( 7, 8,21)(10,28,11) (12,17,13)(14,23,30)(20,25,22)$ $6, 6, 6, 6, 3, 3, 1$ $833280$ $6$ $( 1, 8, 6,21,29, 7)( 2,24, 9)( 3,22,18,20,27,25)( 4,13,26,12,19,17)( 5,15,16) (10,14,28,23,11,30)$ $5, 5, 5, 5, 5, 5, 1$ $666624$ $5$ $( 1, 8,26,13, 5)( 2,18,20,28,14)( 3,22,10,30,24)( 4,17,15,29, 7) ( 6,21,19,12,16)( 9,27,25,11,23)$ $15, 15, 1$ $666624$ $15$ $( 1, 4,16, 8,17, 6,26,15,21,13,29,19, 5, 7,12)( 2,22,23,18,10, 9,20,30,27,28, 24,25,14, 3,11)$ $15, 15, 1$ $666624$ $15$ $( 1,19,15, 8,12,29,26,16, 7,13, 6, 4, 5,21,17)( 2,25,30,18,11,24,20,23, 3,28, 9,22,14,27,10)$ $2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $465$ $2$ $( 1,31)( 2,11)( 3,27)( 4,17)( 8,12)( 9,25)(13,15)(22,28)$ $3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1$ $19840$ $3$ $( 1,15,12)( 2,25, 3)( 6,16,26)( 7,29,21)( 8,31,13)( 9,27,11)(10,20,14) (23,24,30)$ $6, 6, 3, 3, 3, 3, 2, 2, 1, 1, 1$ $416640$ $6$ $( 1, 8,15,31,12,13)( 2,27,25,11, 3, 9)( 4,17)( 6,26,16)( 7,21,29)(10,14,20) (22,28)(23,30,24)$ $7, 7, 7, 7, 1, 1, 1$ $238080$ $7$ $( 1, 4,11,15,22, 3, 8)( 2,13,28,27,12,31,17)( 5,23,30,26,21, 6,14) ( 7,16,29,24,19,20,10)$ $7, 7, 7, 7, 1, 1, 1$ $238080$ $7$ $( 1, 8, 3,22,15,11, 4)( 2,17,31,12,27,28,13)( 5,14, 6,21,26,30,23) ( 7,10,20,19,24,29,16)$ $14, 7, 7, 2, 1$ $714240$ $14$ $( 1,28, 4,27,11,12,15,31,22,17, 3, 2, 8,13)( 5,21,23, 6,30,14,26) ( 7,19,16,20,29,10,24)( 9,25)$ $14, 7, 7, 2, 1$ $714240$ $14$ $( 1,27,15,17, 8,28,11,31, 3,13, 4,12,22, 2)( 5, 6,26,23,14,21,30) ( 7,20,24,16,10,19,29)( 9,25)$ $4, 4, 4, 4, 2, 2, 2, 2, 2, 2, 1, 1, 1$ $78120$ $4$ $( 2,13)( 3,30)( 4, 6)( 5,31,14,25)( 7,28)( 8,17,22,27)(10,26,29,23) (11,18,15,16)(12,20)(19,24)$ $8, 8, 4, 4, 4, 2, 1$ $624960$ $8$ $( 1, 9)( 2,24,13,19)( 3, 7,30,28)( 4,20, 6,12)( 5,26,31,29,14,23,25,10) ( 8,18,17,15,22,16,27,11)$ $4, 4, 4, 4, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1$ $26040$ $4$ $( 1,30, 2, 7)( 3,21,12,24)( 4,28)( 5,17,18,22)( 8,27)( 9,13)(11,31) (15,23,25,29)$ $12, 6, 4, 3, 3, 2, 1$ $833280$ $12$ $( 1,21,25,30,12,29, 2,24,15, 7, 3,23)( 4,28)( 5,22,18,17)( 6,26,16) ( 8, 9,31,27,13,11)(10,14,20)$ $21, 7, 3$ $476160$ $21$ $( 1,30,19,22,16,25,27, 8,21,18, 9,10,13, 2,11,14,20, 4, 7,17,12) ( 3,26,29,15,24,28,31)( 5,23, 6)$ $21, 7, 3$ $476160$ $21$ $( 1, 9,12,18,17,21, 7, 8, 4,27,20,25,14,16,11,22, 2,19,13,30,10) ( 3,15,31,29,28,26,24)( 5,23, 6)$ $4, 4, 4, 4, 4, 4, 2, 2, 1, 1, 1$ $312480$ $4$ $( 1,20, 5,21)( 2, 8, 4, 6)( 7,11)( 9,12,29,14)(10,15,28,27)(13,23) (18,25,26,24)(19,31,22,30)$

## Group invariants

 Order: $9999360=2^{10} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 31$ Cyclic: No Abelian: No Solvable: No GAP id: Data not available
 Character table: Data not available.