Properties

Label 31T10
Degree $31$
Order $9999360$
Cyclic no
Abelian no
Solvable no
Primitive yes
$p$-group no
Group: $\PSL(5,2)$

Learn more about

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

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 TypeSizeOrderRepresentative
$ 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.