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

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.