Properties

Label 31T9
Order \(372000\)
n \(31\)
Cyclic No
Abelian No
Solvable No
Primitive Yes
$p$-group No
Group: $\PSL(3,5)$

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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

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

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.